Not exclusive to only travelers from the U.S, but being American myself, most of the tips for traveling through the States or overseas as an American.
I will try to include anything that will help all foreign travelers that I pick up along the way. (I also work for a visa processing company, so I have some knowledge to share about that process as well).
Most of this will pertain to short visits and tourist visas, not residence in a foreign country.
Make Sure Your Passport is Valid This is a mistake I see most often, because most people check their passport, see that they haven't reached their expiration date and assume they are just fine. NO! Still check your passport. The reason why?
Most airlines will not let you board an international flight with your expiration date within 6 months from when you come back, from your trip. This means, that your trip is for 1 week, your passport must be good for 6 months AND one week.
It is hard, and can be very very costly, depending on whether you live near a passport agency (or embassy if you are a foreign passport holder), or can overnight your passport and required documents into a third-party passport company that can take passport to the agency on your behalf. And don't even think about trying to have it your passport renewed on a weekend (the passport agencies will be closed). So double and triple check your passport before and while you are booking your flight. It can save you a ton of money and headache.