By far, the largest and oldest denomination of Christianity is Roman Catholicism, so they will be first. Roman Catholic theology teaches that when Adam ate the forbidden fruit in the book of Genesis (called "the Fall"), he became an inherently sinful creature with a sinful nature (called "Original Sin") and passed on this sinful nature to his children. This Original Sin infects all of humanity and impairs humanity's ability to have a proper relationship with G-d. Roman Catholic theology teaches that Original Sin doesn't cut off a person completely from G-d, rendering the person spiritually dead, rather it just impairs humanity's ability to have a proper relationship with G-d. Roman Catholicism teaches that G-d wants a proper relationship with all mankind and constantly extends an "invitation" to be in a proper relationship with Him. Grace is granted to mankind through the person of Yeshua/Jesus for the purpose of allowing us to accept G-d's invitation to a proper relationship with Him. Roman Catholicism teaches that Yeshua/Jesus is G-d incarnate and that he alone can grant redemption (salvation) from Original Sin which allows for a proper relationship with G-d. Roman Catholicism essentially teaches that Original Sin leaves such a stain on the soul of a person that unless a person is saved, he will go to Hell after death. Hell is a place of eternal torment for non-Christians that is a punishment for not being saved in Roman Catholic theology, so salvation is viewed as a pardon from this punishment. In Roman Catholicism, Yeshua/Jesus justifies and makes a person holy based on the individual's personal justice and holiness, firstly through the act of baptism. Yeshua/Jesus can work before baptism to grant enough grace through the individual's desire to eventually be baptised, and that is sufficient grace to grant salvation to the individual even before baptism. The reason for this stems from the belief that even though Yeshua/Jesus instituted the sacrament of baptism as a part of the salvific process, he can work independently of his instituted sacraments when he needs to. In the event that a baptised Roman Catholic commits a grave or mortal sin, there are other sacraments that help to bring him back into a saved status and can restore his relationship with G-d. Roman Catholicism teaches that sins are forgiven by confessing them to G-d alone and is not dependent on being a "good Roman Catholic", which enables Christians of other denominations to obtain forgiveness even outside of the Roman Catholic Church. Though it is worth noting that this is a fairly recent teaching in Roman Catholicism and does not reflect the most ancient teachings of the Church. Mortal sins make justification from sin lost completely, even if the individual's faith (the intellectual belief in G-d) is still intact, so a baptised Roman Catholic needs to partake of one of the other sacraments in order to restore his justification according to the ninth canon of the Council of Trent. Roman Catholicism therefore teaches that salvation is the removal of the stain of Original Sin that is inherent in all humanity, that salvation is a pardon from the punishment of Hell, that salvation comes from faith and the act of baptism (or the desire to eventually be baptised), that the justification for sin can be lost through committing mortal sins, and that after committing a mortal sin a Roman Catholic must be reconciled in order obtain salvation again. Roman Catholicism further teaches that salvation can be obtained in other Christian denominations, but that a person must be a part of one of the denominations of Christianity in order to be saved. Salvation guarantees a Christian that he or she will not ever go to Hell, but will ultimately go to heaven and live eternally there with Yeshua/Jesus.
In Eastern Christianity (Eastern Catholicism, the Eastern Orthodoxies, Nestorian Christianity, and Oriental Orthodoxies), salvation is viewed not as being a pardon from punishment as much as it is a healing from the sickness of Original Sin. Eastern Christianity pursues salvation through a process called theosis (drawing closer to G-d and being holy by becoming united with G-d and being an extension of G-d in the world) and forgiveness from individual sins that the person has committed. Eastern Christianity teaches a doctrine called Prevenient Grace, which is the belief that G-d makes the first move to reconcile mankind with Him through His grace, so humanity cannot be saved unless G-d first calls that person to be saved. An individual has free will and can therefore chose to accept or reject G-d's gift of grace, but if the individual accepts this grace, it cannot be said to be of his own accord that he obtained salvation since G-d had to first reach out to the individual and inspire him or her to accept this grace. Eastern Christianity teaches that an individual is guaranteed salvation as long as he or she maintains the faith, and acts of love, and suffers through the various things that happen in life. Therefore, Eastern Christianity teaches that salvation is a healing from the sickness of Original Sin, it comes by faith and acts of love based on that faith (which includes baptism and theosis), and salvation can be lost if the Eastern Christian doesn't maintain his or her faith or acts of love.
Within the Protestant denominations there is division regarding how exactly salvation is attained, and there is even some division regarding what exactly salvation is. Conservative Protestants generally teach more in line with the Roman Catholic opinion that salvation is a pardon from the punishment of Original Sin, while Liberal Protestants generally teach more in line with the Eastern Christian opinion that salvation is a healing from the sickness of Original Sin. The exact method that brings salvation is disputed among Conservative Christians, generally dividing them along the lines of those who adhere to Calvinist doctrine and those who adhere to Arminian doctrine. The Calvinist approach teaches that all mankind is inherently sinful and spiritually dead because of Original Sin, but G-d has chosen an "Elect" few people from before creation to be pardoned from the punishment of Original Sin (the punishment is an eternal Hell), only the Elect can be saved from punishment, that the salvation is impossible to resist (meaning that one of the Elect cannot chose to not be a Christian), and that once a person is saved the salvation is permanent. Calvinist theology teaches that salvation is exclusively because of the Election of G-d and that good works will follow the salvation, there is no possibility of apostasy in Calvinist theology. There is a further division within Calvinist theology over the method used for the Election process, specifically the question of whether the Elect were chosen because they would eventually become Christians or whether the Elect were chosen and that is the reason they became Christians. Arminian theology teaches that Original Sin doesn't render mankind spiritually dead, but spiritually asleep to the point that mankind is unaware of our depravity that came as a result of the Fall. The Arminian approach teaches that nobody is Elect from before creation but they become Elect upon becoming a Christian, anyone can be saved from Original Sin if they choose to become a Christian, the death of Yeshua/Jesus is a sufficient substitutionary atonement to remove Original Sin and his righteousness must be imputed on the individual through choosing to be a Christian, G-d calls all mankind to be saved but individuals can refuse to be saved, but there is no guarantee of salvation unless the Christian maintains his or her faith and good works (meaning there is the chance for apostasy). While most mainstream Conservative Protestant denominations fall into either Calvinist or Arminian theology, there are a few that teach one of many different blends of these two theologies. A minority of Conservative Christians teach that baptism is a requirement for salvation. Liberal Protestants like the Universalists generally teach that salvation is the act of repentance of an individual Christian who uses the death of Yeshua/Jesus as motivation to repent. So Protestantism teaches a variety of different things concerning salvation, but Roman Catholics, Eastern Christians, and Protestants all agree that Original Sin has separated mankind from G-d and only Yeshua/Jesus can remove Original Sin.
The Mormon Church defines salvation according to the teachings of their prophet Joseph Smith, and it's codified in the Doctrines and Covenants and their fourth Article of Faith. So Mormon theology teaches that salvation is dependent on first having faith in Yeshua/Jesus, second by repenting of sin, third by full immersion baptism for the remission of sins, and fourth by the laying on of hands to receive the Holy Ghost. In order to be saved, a Mormon needs all four. The Mormons also teach that all who aren't saved will go to Hell, and that everyone who is saved will live forever in one of the three types of heaven. After a Mormon is saved, he or she has to keep in strict observance of the principles of the Mormon Church in order to maintain his or her salvation, so apostasy is a possibility. While the mainstream Mormon Church (The Church of Jesus Christ of Latter-Day Saints) originally engaged in polygamy according to the teachings of Joseph Smith, they no longer practise it, and it is no longer considered to be a sacrament of the Mormon Church. There are several groups that have split off from the mainstream Mormon Church that do still practise polygamy and consider it to be one of the instituted sacraments of the Gospel, so salvation for these groups also means that men will have more than one wife. These are groups like the Community of Christ, the Fundamentalist Church of Jesus Christ of Latter Day Saints, and at least twenty other similar groups.
The Witnesses teach essentially the same kind of salvation as the Romans Catholic, Eastern Christian, and Protestant denominations in the sense that they believe that Yeshua/Jesus is the only way to obtain salvation, and that mankind has been in a state of depravity since the Fall. For Witnesses, salvation is obtained by faith and keeping the ordinances of the Gospel, though apostasy is possible. The Witnesses are known for teaching that an eternal Hell is a myth, that only a select group of exactly 144,000 Witnesses will go to heaven and live eternally with G-d, and they also teach that everyone who is saved aside from the 144,000 will be resurrected to eternal life on earth. Witnesses teach that those who are not saved simply cease to exist at their death, rather than go to Hell.
Christadelphians teach essentially the same thing as all other denominations of Christianity with regard to the depravity of humanity since the Fall and the role of Yeshua/Jesus of being uniquely able to save mankind from Original Sin. Like the Witnesses, Christadelphians teach that there is no eternal Hell and a person ceases to exist when the person dies. They teach that there is no heaven, but the saved will be resurrected to live eternally on earth with Yeshua/Jesus as the king and Messiah. Christadelphians teach that salvation comes by faith in Yeshua/Jesus, repentance, and full immersion baptism. Additionally, they teach that a person must be "in Christ" to maintain his or her salvation, so apostasy is possible.
Grace Communion International (formerly the Worldwide Church of G-d) was regarded as heretical by the majority of mainstream Christians until recent years for a number of reasons. Since the death of the founder in 1986, the church has abandoned all of its unique doctrines and now affirms a form of Protestant Evangelical Christianity. One of the original objections that mainstream Protestants had to this group was that it required some combination of faith and works to obtain and maintain salvation, but now they are just like any other mainstream Protestant denomination in regards to their theology and soteriology. However, there were some church splits that resulted in several sub-denominations that are still around, but these groups are relatively small.
The United Pentecostals (or Oneness Pentecostals) are a non-Trinitarian, Modalist sub-denomination of the Pentecostal movement of Evangelical Protestants. These Pentecostals teach in line with mainstream Conservative, Arminian Protestants with the notable exception of soteriology. In their soteriology, a person is saved through faith, repentance, full immersion baptism specifically in the name of Yeshua/Jesus (in accordance with the non-Trinitarian baptism formula found in the book of Acts, but in contradiction to the Trinitarian baptism formula found in the gospel of Matthew), and receiving the baptism of the Holy Spirit with the initial sign of speaking in tongues. If any one of these four conditions of salvation is not present, the person is considered to not be saved and will spend eternity in Hell.
The Messianic movement is a sort of Jewish flavoured form of Christianity, and the movement is not a single entity, but a number of different entities, most of which were originally founded as missionary organisations to convert Jews to Christianity. As a result, there is no official distinct soteriology for the Messianic movement. However, because most entities in the Messianic movement were originally started by other denominations of Christianity (mostly Protestant denominations, however Catholic, Mormon, and Oneness Pentecostal groups do exist in the movement), they have a wide range of soteriological opinions that generally match the denominations that the Messianic entity started in. There are individual Messianics that teach that a person must be Messianic and keep the Torah in addition to the other requirements for salvation, but most Messianics hold that faith in Yeshua/Jesus alone is sufficient for salvation and that observance of Torah will naturally come as a result of salvation.
In Christian theology, Yeshua/Jesus never sinned, this is because he was not subject to Original Sin. Christian theology explains that because Adam ate the fruit, all of Adam's descendants inherited a sinful nature, and Yeshua/Jesus was born without a human father. Now, Christian apologists acknowledge that he still had a human mother, but they teach that Original Sin is inherited from the father because Adam ate the forbidden fruit. It seems to me that they forget that Eve ate it first, but that apparently doesn't mean anything regarding the inheritance of Original Sin. In fact, every pastor I've asked about it has given me a blank stare when I bring it up, which tells me that it doesn't have any effect on their theology. So, Yeshua/Jesus was born to a virgin according to Christian theology, which might lead you to ask why his mother (Mary) didn't pass on her Original Sin that she inherited to Yeshua/Jesus. As it turns out, Catholic and Eastern Christian theology has a reasonably good answer for that question. Catholic and Eastern theologians explain that Mary was born without inheriting Original Sin, because of the doctrine of the Immaculate Conception. The doctrine basically says that Mary was kept free of Original Sin and all personal sins that she might have committed because she was chosen before her birth to be the mother of Yeshua/Jesus. This idea is first mentioned in the Protoevangelium of James (also called the Gospel of James), which was written around 145-170 CE. This text is not a part of any Christian Bible, but it did enjoy some popularity among Christians of that time, and it influenced Christian doctrines prior to the Council of Nicea, and some of those doctrines were accepted as valid doctrines at the Council. So because Mary was kept free of Original Sin and (it is taught) she didn't sin, Yeshua/Jesus was born free of Original Sin and was able to never sin. This is why he was uniquely able to atone for sin. The majority of Protestants don't accept the doctrine of Immaculate Conception, so they generally argue that Yeshua/Jesus didn't inherit Original Sin because it's only inherited from the father, though there is absolutely no Scriptural evidence to back up this position even though Protestants pride themselves on basing their beliefs exclusively on their Bibles.
As you can see, there are a wide range of soteriological views expressed within Christianity, some opinions are even mutually exclusive in the sense that you have to belong to a specific denomination in order to be saved. However, all Christians agree that the Fall wounded mankind's relationship with G-d, Original Sin caused our need to be saved to have a proper relationship with G-d, and that only Yeshua/Jesus can restore the relationship with G-d for the individual Christian through his death and resurrection. Most of the finer points are subjects of debate among the denominations of Christianity, but they all agree on these. With all these opinions on salvation and the need for Yeshua/Jesus to be saved, it begs the question of what G-d's opinion on the matter is. Isn't it best to just see what His opinion is since His opinion is the only One that matters?
While the concept of needing salvation is a core doctrine of Christianity, the idea is completely foreign to Judaism. Part of the reason for this is there is no concept of Original Sin in Judaism, as the prophet Ezekiel put it (18:20), everyone is held accountable for their own sins. The Torah teaches (Deuteronomy 24:16) that sin cannot be transmitted from one person to another and one person cannot take the penalty for a sin he didn't commit. Automatically, Yeshua/Jesus taking the punishment for anyone else's sin is ruled out by the Torah. One of Christianity's core doctrines doesn't go down with just one passage in Scripture that contradicts it, and I would never expect it to. Christians generally have to argue that Yeshua/Jesus was different though, because he was also divine. This also doesn't work for the Torah because Numbers 23:19 (and a few other passages) specifically says that G-d cannot be a man. Christians will counter this by saying that G-d had not been incarnated in the person of Yeshua/Jesus yet, so it's obvious that G-d was not a man at the time that Numbers was written. The biggest problem with this argument for Judaism is that the Torah is eternal and what it says cannot be contradicted, but I'm not going to belabour that problem because Christians generally won't accept that as a valid argument. But just ask yourself if you would be willing to play a game where the rules change every so often and you aren't told when they change or what they are changed to, and that's basically what Christianity has done with the Torah. A better argument in favour of Numbers 23:19 is that the gospel of John chapter 1, verse 1 says that the Word (Yeshua/Jesus) was G-d "in the beginning". The plain reading of this is that the person of "god the son" was G-d before the earth was created. In fact, the Nicean Creed, the statement of faith that the bishops of the entire Church agreed was what they believed in 325 CE, states that G-d has eternally existed in three distinct persons (the doctrine of the Trinity), which would make "G-d" a sort of title for a council of sorts that has three persons on the council. From my own research, it appears that the doctrine of the Trinity teaches that Yeshua/Jesus is not the same person as the Father or the Holy Ghost/Spirit, and that he is not G-d by himself, but needs the other two members of the Trinity in order to be G-d. But to be fair, I might have misunderstood that. So if Yeshua/Jesus has eternally been G-d, and he is G-d incarnate, then G-d has always been a man, which directly contradicts Numbers 23:19. So Yeshua/Jesus wasn't G-d, which means that the main reason that he would have been able to take the punishment for humanity's sin is now gone.
According to the Torah, there is no such thing as Original Sin that's inherited by children of the sinners, so there's no need for salvation in the Christian usage of the word. "The Fall", therefore, also doesn't exist and mankind can enjoy a proper relationship with G-d simply by obeying, worshiping, and praying to Him alone. By "Him alone", I mean that Yeshua/Jesus isn't a part of a triune godhead, and it's improper and outright idolatrous to worship or pray to him. I apologise if that's offensive to you, but I'm sure that you would rather read the truth than for me to tiptoe around the subject and be politically correct. You may be wondering what Scriptural evidence I have that shows that Original Sin doesn't exist, aside from the obvious fact that G-d never says it does in the entire Jewish Scriptures. After Adam and Eve ate the forbidden fruit, but before Cain killed his brother, there was an incident that shows that humanity wasn't depraved like all Christianity teaches. If you'll look at Genesis 4:6-7, you'll see G-d speaking to Cain. G-d tells Cain that he can be forgiven if he just "improves himself (repents)" and then tells him that sin is waiting at the door for him, and sin wants to rule him, but he can become the master over sin. If Cain had inherited the stain of Original Sin from his father Adam and that Original Sin was so strong that Cain was a slave to it (as all Christianity teaches, based on Romans), how could Cain be the master of his sin? Ephesians 2:1 and Colossians 2:13 both say that we are dead in our sins, and a dead man can't be a master of anything. John 8:34 and Romans 6:20 both teach that we are slaves to sin, yet G-d says to Cain that we have the power to make sin our slave. If we are slaves to sin, then we cannot be masters over our sin. If we are dead in our sins, then we can't do anything to combat our sin. But G-d says that's pure and total nonsense, not only are we not dead in sin or enslaved to sin, but G-d told Cain that sin is waiting at the door for him, which means that sin doesn't even have the power to enter our lives unless we allow it in. So the idea that we can inherit the sin of our ancestors is directly refuted by G-d Himself in His Torah. That's G-d's opinion on Original Sin, that it doesn't exist and that we have the ability to be the masters of sin if we are willing to obey G-d alone.
You might be wondering if salvation is mentioned anywhere in Jewish Scripture. After all, if it's important, G-d surely wouldn't wait until Yeshua/Jesus to tell us about it. As it turns out, salvation is mentioned a surprisingly high number of times in the Jewish Scriptures. For example, Exodus 14:13, Exodus 15:2, Deuteronomy 32:15, Psalm 3:8, Psalm 9:14, Psalm 13:5, Psalm 62:2, and many other passages (especially in the Psalms) all speak about salvation. I recommend getting a concordance and looking up the word "salvation" and comparing the salvation of the Jewish Scriptures to the salvation of the New Testament. If you notice, every time the word "salvation" is used in the Jewish Scriptures, it always refers to being saved from a physical threat and it never uses it in the same way Christians and the New Testament do. Wouldn't it be logical that if we had a need to be saved from our sins or the punishment of our sins that G-d would have mentioned it before Yeshua/Jesus so that the Jewish people would recognise that Yeshua/Jesus was that salvation? So why does G-d never say we need to be saved in the sense that Christianity uses the word? The answer is plainly obvious, the salvation that Christianity offers is not the salvation of G-d. It seems to me that Christianity invented the issue of Original Sin and then offered their guy as the only cure for the illness. So why do we need Yeshua/Jesus? To put it bluntly, we don't.