I suppose it goes without saying, you need a passport to travel. Nearly every country will deny you entry without a passport that is valid for at least 6 months past your date of entry. You should actually make sure it is valid for at least a year to avoid any issues. Mine was fine, but I didn’t have enough pages so I had more added.
In many parts of the world, immigration officials are stamp happy and they want plenty of room to stamp away on your passport. They get really irritated when there’s not enough space. Before I had new pages added to my passport, I was scolded by Brazilian officials on a couple of occasions and almost wasn’t allowed to depart the country because I literally had only one spot left on my passport to stamp (and it doesn’t help that Brazilian stamps are larger than most stamps). The Italians, on the other hand, stamped whereever they felt like it without regard to whether there was a stamp there already or not.
Many countries require a page or two for visas, so it’s a good idea to have plenty of room in your passport. It’s best to acquire your visas before you go. It makes no difference if you do it yourself or have an agency do it (the agency is just a bit more convenient, but also more costly as they usually charge around $50 per visa for their services). Getting visas at the border can be time-consuming (in some cases, weeks) and expensive. In certain circumstances, it isn’t even possible—you are required to do it in you home country.
Many, though by no means all, counties allow Americans 30 to 90 days when arriving, though every country is different. Thailand is 30 days if you arrive by air and is actually a visa waiver (they still stamp the passport, it is just not an actual visa). I could have gotten an actual visa but decided I would be doing enough regional travel that would allow me to be able to get new stamps every month as needed. India, on the other hand, requires a full visa so I took care of that in advance. I thought about getting my Chinese visa now, but decided to wait.
If you are not sure about visa requirements, check out the country’s embassy and the State Department’s website. They have good information about each country and what is required. They have an iPhone app that you might want to check out as well. They also have a program called Smart Traveler Enrollment Program (STEP) that allows you to register with the State Department for where you will be. This is good in case something happens while you are traveling.
Visa requirements change constantly, so always double-check before you go. Remember, you will be a visitor to the country in question. Just because you have a visa does not guarantee entry, nor does it guarantee entry for the full amount of time. Once when going to Guatemala I was only given 8 days even though 30+ days is more standard. I had to go through a lot of hassle to increase the length.
Be polite and request what you want before they stamp your passport. If you don’t get what you want, remain polite and leave when you are supposed to. Overstaying a visa is not a good situation (it will result in you being fined and barred from the country in the future). This is because they think you sold your passport or purposefully lost it to hide the fact that you overstayed your visa.
Make sure you have a copy of your passport with you while you are traveling. This really helps if you lose your passport. I also scan mine and keep a copy on Dropbox and in my email for easy recovery. It’s a good idea to also have pictures of your visas. This will make things much easier for you should you need to replace those if you lost your passport.