Christianity has a strong oral tradition, though it varies in content and importance from denomination to denomination, church to church, Christian to Christian. I firmly believe that in some areas and on some issues, the oral tradition actually takes precedence over the written tradition - the written tradition being, of course, the Bible. In this case, however, the written tradition retains its dominance. I think that's mostly due to the letters of the Apostle Paul, but I'm really not sure - it could just be one of those things that's worded so compellingly and unmistakably that the oral tradition has trouble changing it.
This is kind of a problem, because it keeps the Christian approach to evangelism firmly rooted in a situation that has been out of date for sixteen hundred years.
The Christianity of the Apostles was very similar to a doomsday cult: Christians were an extreme minority, and they believed that the Messiah was coming back within their lifetimes to judge the living and the dead. Explaining the Good News was urgent, vital, and utterly necessary. The people that the early Christians were evangelizing had never heard about it before. Perhaps equally important, the Gospel was a deeply subversive story. There's only one God? And His son was born to an unmarried woman, in a feeding trough in a barn, and spent his life as an itinerant preacher and sometime healer? And this holy son's great achievement, his one truly divine accomplishment, was to be tortured and die in a particularly horrible and ignominious fashion?
...It's a wonder that Christianity made it off the ground at all.
But the most successful stories often are subversive at heart, and a millenia and half later I have to explain why those ideas might have been a bit... unusual. Christianity isn't just widespread; in most places, it's the default. In some places, it's part of the definition of "normal" - if you aren't Christian, you're abnormal. Nowadays, the idea that most people in Western nations have never heard of Jesus is, frankly, silly. Christianity is so dominant that here in the 'States, even the adherents of other religions understand its broad outlines. Yet Christianity clings to this view of itself as a minority religion (and frequently persecuted at that), in no small part because the written tradition - the Bible - says that it must be so. In much the same way, Christian doctrine clings to the idea that non-Christians must never have heard the Good News of Jesus, or - failing that - must not have understood it.
The idea that someone could know, and understand, but still disagree escapes far too many people. The fact that these assumptions lead to Christians mainly evangelizing other Christians is actually kind of funny. Keep these thoughts in mind; we'll likely be coming back to them.
This leads me to two initial pieces of advice, if you wish to share the good news without giving offense:
- Do us the courtesy of assuming that we're already familiar with the broad outlines of Christianity. Odds are, you aren't the first to share the Good News with us.
- When a Christian starts making generalizations about why believers don't believe, cover your ears and walk away. The odds that they have anything useful to say are so close to zero as gives no odds.
Next up: Finding a workable goal.
 Monotheism in the Roman Empire was... unusual, and it was probably a big part of the reason why early Christians were persecuted. Accidental Historian has some comments about this.