The central problem of Christianity, from the point of view of non-Christians, is who Jesus was. CS Lewis famously summed it up in Mere Christianity by saying that either Jesus was a lunatic who thought that he was God when he wasn't, or he was a liar, who knew he wasn't God but claimed to be, or that he was Lord - he claimed to be God and he really was God. Lewis then goes on to show that the evidence is strongly against Jesus being either a liar or a lunatic, and therefore it is highly likely that he is Lord.
Various attempts have been made to get out of this. Some people try saying that those aren't the only three possibilities, and try to concoct a fourth, usually by mixing the ideas of liar and lunatic, which they don't notice still falls foul of the same evidence. An altogether cleverer way out is to question whether Jesus actually claimed to be God at all. The Bible clearly portrays him as doing so, but what if there is a difference between the Jesus of faith, as presented in the Bible, and the Historical Jesus - Jesus as he actually was?
(And yes, before people get penickety, I know that the Historical Jesus movement started a long time before CS Lewis, and that some of them (e.g. Borg) are coming from somewhere different to my description above. Borg, for instance, is trying to present a Christianity that fits in with his worldview where God can't act at all in history. I discuss that issue here.)
The evidence that they focus on tends to be things like the difference between different accounts in the gospels. If the accounts are too similar, they say they are copied from each other, and if the accounts are too different, they suggest it is because the writers are making things up. This is especially true with John, because John is very different to Matthew, Mark and Luke in lots of ways, so some people think it is mostly an invention, and that Jesus didn't say most of the things in John. (Incidentally, some good has come out of this too, as it has made people look more carefully at why the gospel writers structured things the way they did, and so helped us to see their emphases, main points, etc.)
I want to suggest that a lot of the questions that are being asked are actually irrelevant, that Jesus only said two things he is recorded as saying in the gospels, and that we can tell that the gospels provide true accounts of Jesus.
A lot of this is because of the simple problems of translation. The Bible we read today is in English. The gospels - the bits directly about Jesus - were originally written in Greek (well, some people argue that Matthew wasn't, but the earliest copies we have are in Greek). But Jesus almost certainly spoke Aramaic and Hebrew most of the time. So we know that what we have in the gospels is at best a translation of what Jesus said. There are two exceptions, where Jesus' words in Aramaic are recorded - "talitha koum" (Little girl, get up) and "eloi, eloi lama sabacthani" (My God, My God, why have you forsaken me?).
But it gets more complicated than that. Greek doesn't have a neat distinction between direct speech and indirect speech - you can't tell the difference between "He told me to get up" and "He told me: 'Get Up!'" So we can't tell whether what we have are the words of Jesus neatly translated into Greek, or whether it is the apostles reporting what Jesus said indirectly.
And it gets even more complicated. Greek and Aramaic aren't very similar as languages. It isn't possible just to translate straight from one into the other and keep the sense the same. It's like the problem of translating the Bible into English. Some people translate literally word for word and lose the flow of what is said and sometimes leave it incomprehensible. Other people translate so that it is the same meaning, but it's said quite differently. We don't know exactly how the people who wrote the gospels went about translating Jesus' words into Greek. I've suggested that Matthew, Mark and Luke may have gone for a more literal translation, John may have gone for a translation that aimed to convey the same sense, but not necessarily translating literally. (Of course, John might have been literally translating what Jesus said - I don't think we can know this side of heaven.)
Which rather leaves us with a problem. What we have in the Bible is a translation of either what Jesus said, or what he meant, quite possibly put into the authors' own words. How can we know they are reporting it accurately?
I think the answer to that is fairly simple, and often missed by the Historical Jesus scholars. The people who wrote it clearly believed that it was true. The people they wrote it for clearly believed it was true. They quoted what the apostles described Jesus as saying as being what Jesus said. And the people they wrote for included people who had known Jesus. The early church, while some of the apostles were still around, regarded the gospels as faithful records of what Jesus had said and meant. And they were in a position to know.
But that doesn't mean they were gullible. The gospels were all written before AD100. After then, other accounts of what Jesus said and did were written (and the Gospel of Thomas might have been earlier). A few people were taken in by them, but the church as a whole rejected them because they weren't saying what Jesus said - they were saying what the authors wanted Jesus to have said. They could tell the difference, and they did.
We can tell that the gospels provide a reliable account of the historical Jesus, even if it is in translation, because the people who were in a position to know what Jesus did say and do agreed with the gospels. And these were not academics or people just along for the ride. These were people who staked their lives on what Jesus had said and done, repeatedly, and in most cases were killed for it. Whether the gospels record something very close to what Jesus actually said and did, or whether they provide an interpretation of what Jesus said and did, they're still true.
And so the problem for non-Christians remains. The people who were following Jesus were willing to stake their lives not only on Jesus claiming to be God, but on Jesus actually being God. Who was he? Was he mad, bad or God?