#1) Most of the world's people are friendly and decent.
Except for the French*.
Some stereotypes really hold up, but on average, most of the people I've met around the world are extremely polite, friendly and helpful. They are generally interested in why I chose to visit their home. They are eager to assist if it's obvious I'm lost or in trouble. They'll go out of their way to try to make sure I have a good stay in their country. And, contrary to what most Americans tend to think (see #3 below), they often don't know much about the United States (or necessarily care).
Don't be convinced before leaving that "everyone there is _______". Show a modicum of respect to people and their culture and you'll be blown away by what you get back. Try picking up a little of the local language. Just learning how to say "thank you" can make a huge impact.
* Sorry, I couldn't resist. To be fair, France is like everywhere else: most people are decent. It's just that France has a particularly large proportion of bad apples that give the place a well-deserved reputation. I've met a lot of wonderful people in France, but also a disproportionate number of a**holes (not travelers generally, but residents of France). This isn't based on a single trip nor is it restricted to Paris. Almost every non-French local in Europe agreed with me on this one.
#2) Most places are as safe (or safer) than home.
I remember confessing to my mother recently that I had a big night out in Budapest and stumbled back to my apartment at dawn. Her reaction was: "But don't you worry about being drunk in a foreign country?"
Ha ha, not at all mom! I've never felt so safe!
The only place I've been violently mugged was in my home city of San Francisco. Many of the people I know there have been robbed at gunpoint, and on more than one occasion there were shootings in my neighborhood.
In one incident just a block away from my apartment (Dolores Park), a man was shot five times and somehow escaped, only to collapse about 10 meters from our front door. You can still see the blood stains on the sidewalk.
Turns out we actually live in a pretty dangerous country.
In over 365 days on the road, staying mostly in dormitory-style hostels and traveling through several countries considered 'high-risk,' the only incident I had was an iPhone stolen out of my pocket on the metro in Medellin, Colombia. I didn't even notice and deserved it for waiving the damn thing around in the wrong part of town. Most people think that in a place like Colombia you'll still get kidnapped or knocked off by a motorcycle assassin, but that's not true. According to the locals I talked to (who grew up there), things have been safer there for at least 10 years.
Caveat: This doesn't give you a license to be stupid, and some places really warrant respect. Guatemala and Honduras, where there are major drug wars going on (and the Peace Corps recently pulled all of their volunteers), or Quito, Ecuador, where everyone I talked to had been robbed, are reasonably dangerous (I had no trouble in any of them).
In reality, based on the sort of mindless binge-drinking that happens in most travel hot spots, you'd expect travelers to get knocked off a lot more often. But if you pay attention and don't do anything stupid, you'll be fine.
#3) You can travel long-term for the price of rent and a round of drinks back home
My favorite question from friends at home has been: "how the hell are you still traveling?"
Well, for what you spent at lunch I can live on for a whole day in Indonesia. That's all there is to it.
