Yes, there are contradictions in the Gospels. It is common for atheists to compile a list of contradictions and other difficulties, and for a Christian to respond. See here and here for a good example of the back-and-forth. Some Christians will try to defend each and every alleged contradiction as not being a contradiction at all, since they have committed themselves to the proposition that scripture is free from error (I have not). I think that a fair-minded reading of any of these debates will result in a split decision of sorts, where some of the critics’ claims are off-base, but nonetheless there remain some difficulties in the Gospels that cannot be plausibly harmonized. Some of the details are either mistakes, or intentional dramatizations.
Does the presence of mistakes discredit the historicity of the Gospels? I think not. Consider that in recent times there have been excellent films made about famous figures such as Mahatma Gandhi, Queen Elizabeth, Ray Charles, Johnny Cash, and Oskar Schindler. Each of these films has been critically acclaimed and generally agreed to be based on a true story, and faithful to the major events and the character of the subjects. However, when subjected to close inspection, one finds numerous inconsistencies and errors, just within any one given film. See “goofs” pages on IMDB for these films here, here, here, here, and here. It seems pretty clear that the filmmakers made errors. Some of them were simply mistakes, others were dramatizations or re-ordering of events in order to make the story flow more smoothly. Not every word spoken by every actor was historically spoken by their subject verbatim, although many were. However, these departures from a literal presentation of fact did not make these films works of fiction. They are clearly efforts to faithfully present historical events and convey the character and ideas of their subjects. In fact, they made the films more memorable, more widely viewed, and more powerful than if they had been done as documentaries.
I would suggest that the Gospels are best seen in the same light. The ancient world had very few writers whom we would today call journalists or professional historians. Access to highly accurate information was in shorter supply. This difficulty applies not just to Jesus but to any ancient figure. Every narrative that was considered worth preserving from ancient times contained not only facts, but good storytelling. Then, as now, we can form judgments based on the stories we hear whether to classify them best as fiction, propaganda, “based on a true story”, or literal historical events. Each person will do this differently based on whatever presuppositions they bring to the process. If you are committed to the idea that miracles are absolutely impossible, you would consider the Gospels to be at their root based on a hoax, and build your interpretation around that. If you believe in God to begin with, on other grounds, you might be open to reading the Gospels as honest accounts of events that actually happened, albeit presented in a dramatic way. There are sound reasons to find the stories of Jesus more credible those of other religious traditions (see here and here). Ultimately, though, it comes down to whether you recognize His voice.
Having four Gospels handed down from the first-century Church, written somewhat independently of one another (but also likely referencing very early works now lost to us), we are in a position to compare them. Major events happening in all four Gospels (such as the resurrection, or the miracle of loaves and fishes feeding a crowd of thousands) are very likely to be authentic. Many of the parables are easy to memorize and retell almost verbatim, so the version we had are probably very close to how Jesus originally told them. Longer speeches by Jesus of a more theological nature (such as are primarily found in John) are probably not word for word recollections, but rather an organized presentation of what the community of each Gospel writer believed at the time the Gospel was written. Where these ideas overlap across Gospels, they probably go back to the very beginning. Where a specific Gospel writer appears to be putting his personal spin on things, it might be just that. Conscience, common sense, an understanding of the history of that time, and comparison with other parts of the New Testament (e.g. the epistles of Paul and John) are the tools I use in trying to build as accurate a picture of the Lord’s life and teachings as I can. Nonetheless, anyone familiar with the bare highlights of what Jesus did and taught probably has everything they need to know. God’s full truth is beyond our full comprehension, but what we need to know of it here is simple enough for children to understand, and that’s the correct spirit in which to approach it.
Incidentally, one might ask why Christians use four Gospels, rather than just one, or a large number. The reason we don’t have just one is that Jesus Himself did not write a Gospel. I think this was intentional. God’s purposes are mysterious, but if I might speculate, I would think that His plan for spreading His message to us was primarily intended to be person-to-person. The written texts were tools that the early Christian communities found useful in standardizing, and preserving, and spreading the message. This was all well and good, and probably necessary to prevent the Church from splintering off onto wild tangents having nothing to do with the original message (and if this was the intent, it has of course only been partially successful). However, it should be kept in mind that most Christians didn’t have access to written Bibles for at least fifteen centuries (and in some places, a lot longer). The idea of getting to know God primarily through studying a written text is something Jesus apparently frowns upon, as suggested specifically by John 5:37-40 and more generally from the whole thrust of Jesus’s ministry, away from ritual and legalistic analysis of scripture, and towards an emphasis on love, expressed person to person.
The reason Christians canonized just four Gospels (Matthew, Mark, Luke, and John) is not for lack of their being many more; a list of 84 “Gospels” can be found here. We rely on the judgment of the early Church as to which of the purported Gospels had an early and genuine origin, either as eyewitnesses themselves or close companions of eyewitnesses. Contrary to the impression left on many by the influential (yet fictional) novel The Da Vinci Code, for the most part these judgments are not in conflict with those of modern historians. The four Gospels are widely recognized to be early accounts written within the lifetime of those who knew Jesus. The “Gospels” of the second and third centuries may contain fragments of early material, but for the most part were inventions of other communities following a faith different from that of the early church. Probably the only Gospel aside from the canonical ones for which there is a good case for an early origin is the Gospel of Thomas. Scholars have a wide variety of opinions about it. I’ve read it and concluded that perhaps it contains some genuine sayings of Jesus. Still, it wasn’t as organized as the canonical four, and it didn’t make a big impact on my thinking.