Sorting through the mountains of information regarding the historical reliability of the gospels is difficult and daunting. The subjects are many, including textual criticism, history, comparative study between the gospels, the nature of memory, oral storytelling, the question of genre, etc. How does one pick out what is important? How does one approach the reams of material in each of these categories?
Note: This is not an academic paper, and thus is not footnoted. For more reading I would suggest either E. P. Sanders (The Historical Figure of Jesus) or Geza Vermes (Jesus).
Thankfully, a whole lot of this information can be simply put aside once a few things are acknowledged. Starting with textual criticism, and without getting into details, there is good reason to think our copies of the Greek text of the gospels are very good. This doesn't mean the claims of the gospels are accurate, but it does mean that we know the texts of the gospels were not greatly corrupted in the copying process. Having recognized this, we don't need to worry about textual criticism for the question at hand.
The study of memory has thrown doubt on how solid our memories really are, but there is not to much to be done about it. Having multiple sources is good because it reduces the chances of memory being wrong. Studies on oral storytelling have been brought into the question of the gospels reliability, but they really don't add too much. We don't have access to the disciples oral culture, so studies of modern oral storytelling cultures are interesting, but far from conclusive. Whatever genre the gospels are, early Christians seemed to think that they reflected real events. We can and should ask the question of whether they are accurate or not even if there was not an expectation of modern historical accuracy.
So, the texts are good enough. Memory is what it is, and we can and should ask if the events in the gospels really happened, but perhaps without being too picky about detail. And this is exactly where we find some big problems.
Comparative study of the gospels demonstrates substantial agreement between Matthew, Mark, and Luke. Matthew and Luke seem to have used Mark as a source, and so usually when they agree with Mark they agree with each other. This agreement doesn't mean historical reliability because Mark might have been wrong. Where they don't have Mark to follow, we see some serious differences between Matthew and Luke.
The birth narratives of Matthew and Luke are so different as to be contradictory. Matthew has Joseph and Mary living in Bethlehem one to two years after Jesus' birth (when the wise men visit), and then avoiding Judea at all costs by first fleeing to Egypt and then moving to Nazareth. Luke has Joseph and Mary move back to Nazareth after Mary's purification and then visit the temple every year in Jerusalem. So if Luke is right then the Wise Men visited a Jesus who wasn't there. And Mary and Joseph didn't avoid Judea at all. Matthew and Luke seem to be working with different sources that don't agree with another, and for which there is no easy way to make them fit together.
The resurrection accounts are all over the place. There is agreement that Jesus was dead and buried, but outside of that the details are a forest of minor disagreement. There might be a way to harmonize them save for one substantial difference between Matthew and Luke. Matthew has Jesus commanding the disciples to go to Galilee, and Luke has Jesus commanding the disciples to stay in Jerusalem. Whether or not the disciples stayed in Jerusalem or went to Galilee Jesus is reported as giving two contradictory commands.
So for two of the most important historical questions for Christian belief and practice, the birth of Jesus and his resurrection, there is not agreement between the sources. And we don't have any other sources. So Matthew and Luke cannot both be right on Jesus' birth. One of them might be right, but we couldn't know because we don't have any other sources. In fact they might both be wrong. Comparative study of the gospels alerts us that on some important events the sources don't agree, and therefore we don't have unified testimony in the way we would like. I have given only two examples, but there are more.
When we dive into historical study as a whole, the gospels are sometimes out of step with what we know about the ancient world. Luke has the census of Quirinius happening when Herod the Great ruled, but these events were ten years apart. Some of the details about Jesus' trial seem strange and out of step with what we would expect. So the gospels don't always agree with what we know about the rest of the ancient world.
These observations don't mean the gospels are wrong about everything. Possibility the gospels get many things right. What these observations mean is that we don't know. What we know is that the gospels are sometimes wrong and not always in agreement with each other. In this they are much like any other historical documents. But given that we don't have any other sources of Jesus' life, and that gospels are at best a second hand telling of the events of Jesus' life, and don't even know for sure who wrote them, pronouncing them historically reliable seems a stretch.
Again, this isn't to say they are wrong about everything. Or right. We just don't know. Certainly we can read them as we would other historical sources and give them the benefit of the doubt unless we know better. But this isn't how Christians read the gospels or use the gospels. Christians make the claim that the gospels are true. But this can't be. Jesus can't have both told his disciples to go to Galilee and stay in Jerusalem. And both couldn't have happened the way the gospel writers describe the events. Granted, this one disagreement doesn't make the gospels wholly unreliable, but it does prove that you cannot assume they are accurate and reliable.
Christians make claims that need the historical reliability that the gospels are unable to give us. The Christian message is that Jesus died for us and rose from the dead to save us from eternal punishment. Christians claim that Jesus is the Lord of all people, and that all people must believe in him and bow the knee or face eternal torture in hell. This message demands a total life change from all people and threatens terrible consequences if it is ignored.
But then we don't have documents that we know are historically reliable. How can eternal punishment be threatened when we don't even know for certain about Jesus' life?
Some might say that there is enough. That the gospels agree on the main points. This amounts to saying even though I know at least parts of the gospels must be historically unreliable because of their disagreement, that where ever the gospels agree it must be true. This is first a non sequitor: all the gospels might be wrong about an event they agree on. The truth is that we don't know. Furthermore if only the agreement of the gospels is taken as true, then there is not much left to work with besides a skeleton of facts that doesn't lead to much.
Basic agreement with substantial disagreement in the details seems hardly reassuring in light of the seriousness of Christian claims. The virgin birth, for example, is important for Christian theology. But Luke and Matthew tell two entirely different stories about it. These stories do not directly support the other even if they are compatible. They both could be true, they both could be false, only one of them might be true. We don't know. That Luke and Matthew agree that Mary was a virgin doesn't mean they are historically reliable on that particular fact! And the doctrine of the Incarnation, which is central to Christianity, depends on Jesus being born of a virgin and the Holy Spirit.
In short, we know that the gospels are occasionally not historically reliable, and we don't know know where they are historically reliable. This directly undermines claims of Christianity to be true, based on historical happenings. The truth is, we don't know for sure what happened with Jesus because we don't know if or where the gospels are historically reliable.