First, let me tell you that there are sometimes recommendations specific to a certain country, specific banks and specific cards, so be sure to do a little research before you go for some additional guidance, but here are my recommendations based on my own experiences.
Forget about travelers checks. Some of my clients ask me about travelers checks, and the truth is, many countries won't even accept them any more, and they can be difficult (and costly) to get. It's best just not to bother.
Don't worry about changing money before you go. It's just another unnecessary hassle before your trip, and exchange rates are going to be bad. Just wait until you're at your destination.
Avoid exchanging money altogether. As much as possible, it usually makes sense to use a credit or debit card rather than cash. In some countries (like Iceland), you'll be hard pressed to find an establishment that doesn't accept plastic. In other places, you will need some cash for smaller shops or street markets, but you'll still be able to use your card (and avoid awful exchange rates) most of the time. Just double-check to make sure that your card doesn't charge a foreign transaction fee.
When you exchange money, use the ATM. Most ATMs will have a small(ish) charge for foreign cards, but you won't pay the commission and fees charged at a lot of cash exchange businesses AND you won't have to travel with cash to be exchanged when you arrive. And some banks will actually reimburse you for your foreign ATM fees, so it's good to do some research before you go!
Double-check that your card has no foreign transaction fees. I mentioned this above, but it's really important. If you're using your card for most things, just be sure that it's not charging a fee for every time you use it internationally. As you might expect, this can seriously add up.
Pay in local currency (not US dollars). Some places you'll visit will accept US dollars or will even charge your card in US dollars for you. Most of the time, this is a bad idea, and paying in local currency is the way to go. You'll usually pay less with fewer fees if you choose the local currency, but, again, this can vary based on your specific card, so make sure you know your own bank's rules.
Take a minute to familiarize yourself with the local currency. Get an idea of which coins and bills are for which amounts so that you avoid flashing a whole stack of bills and dumping out a pocketful of change when you pay. Also, an American who's used to ignoring the jingly sort of change, I'm sometimes surprised at how valuable coins can be in other countries. (For example, in Japan they use a ¥500 coin that is worth about $4.50!)
Double-check the type of card that you're using. Visa and MasterCard are pretty widely accepted, but you are sometimes hard-pressed to find places that will accept a Discover or American Express card, so be sure those aren't the only cards you're traveling with. Again, check what's normal for your destination!
Those are my experiences and recommendations, but, as I've mentioned, it's vital to know your own card, your own destination and your own buying habits to make the best choices for you!