"Question: Wasn't the New Testament written centuries after the death of Jesus, handed down as oral tradition from one generation to another? So how can we know what actually happened?"
This idea, which I have heard often, is entirely false. As it turns out, the bulk of the New Testament was written by eyewitnesses, or their close associates, and their words were penned within 15-30 years of the events they describe.
First, with respect to the four accounts of Jesus' life, known as the "gospels", these are believed to have been written beginning about 30 years or so after the death of Jesus. Now that's long enough that some details might get hazy, or that the stories about Jesus might get rearranged out of strict chronological order, but much too soon for outright lies to gain any traction.
The apostles' original manuscripts are lost to history, but we have early testimony about them, suggesting that by mid-second century copies of these writings were floating around and being used in the churches. There may even be a fragment of the Gospel of Matthew, known as the "Magdalen papyrus", dating to the 60s AD (this claim is not universally accepted). We have many later copies and fragments of the gospel manuscripts, and it turns out that they all are virtually identical to each other, with very few additions or changes to suggest an evolution of the story over time. Most differences from one document to another consist of changes in punctuation or slight changes in word order. Thus, we can be fairly certain that the gospels we have contain the words that were originally written by the eyewitnesses, not fantastic stories inserted by their grandchildren. If you get a study bible, it will often highlight alternate wording, or bracket off the sections that may be later additions (again, there aren't many such passages).
The letters (or "epistles") of Paul were written even earlier, some of them within 15 years after the crucifixion of Jesus. Again thousands of manuscripts of Paul's letters exist, with few changes or additions being made over time. As with the gospels, we can be confident that today's Bible contains the actual words of Paul, not stuff made up and inserted by later generations of believers.
So, in sum, the New Testament contains the actual testimony of the people who were alive and present during those remarkable events. Their story spread very quickly and almost immediately won thousands of converts. I might just leave this issue here with a couple of questions for you to ponder: What would make thousands of people who were raised as Jews, to whom it would be an outrageous blasphemy for someone to claim to be God, suddenly and mysteriously to turn around and worship Jesus as God. Also, would a bunch of people conspiring to invent a story be willing to suffer all kinds of torments, and die all sorts of gruesome deaths, if they knew at the outset that it was all untrue?