I've been traveling internationally for years. Most of the comments above are essentially correct. However, IMHO, a better approach is to purchase an "unlocked" GSM quad-band phone while you are still in the US. Why quad band? If you go to another country that is not supported on a frequency supported by your phone, you're out of luck. Here's the deal:
There are 2 frequency "bands" used by GSM services in the US. However, 2 entirely different bands are used by GSM services
elsewhere in the world. If possible, you will want to purchase a phone that can work on all four bands - these are quad-band phone. If you purchase a "tri-band" phone as a second best choice, be careful which three bands your phone has. It simply depends on where you travel. Again, cover the possibilities, and purchase a quad-band phone upfront. If this sounds complicated, it's not. The only issue here is that not all countries are operating on the same band / frequency.
The GSM community has a website that will tell you what countries is operating on what frequency. If you buy a quad-band phone, it won't matter.
http://www.gsmworld.com/roaming/gsminfo/index.shtml.
So...researching the above information for where you'll be traveling, as a suggestion, purchase an
unlocked GSM phone from either an eBay seller or Amazon.com. The phone I like best I purchased on eBay for about $30. (Or purchase from a seller you feel comfortable with. Some Mom and Pop local stores...if you can find one...may have a new or used phone that is either already unlocked, or they will unlock for you. Generally, an AT&T or T-Mobile store will not sell you an unlocked phone. But, you can check)
If possible, purchase a
Sim Card in the US. Sim Cards don't give you "minutes", but they are necessary for operation of a GSM phone. My experience, most eBay sellers don't normally provide)
With phone in hand, get your hotel to recommend a nearby cellular store.
(Yes, there are always vendors in the airport, but do you want a potentially expensive ride back to the airport if you encounter a problem?). With a fully charged phone in hand, show the store your phone and request a prepaid card with the appropriate minutes you require. You will get a phone number for your phone. Ask the store for help in terms of how to call your number, both locally, and from the US (or other country). If possible, have them call your now working phone. You also have the number of the store in case you have questions.
(Yes, these stores will also sell you a phone, and there may be something to be said for piece of mind if you have a problem or need support. In my opinion, expect to pay $100-$300 for a phone, sim card, and a prepaid minutes card. You decide)
Lastly...my experiences. I have never bought a phone thru Amazon, but lots of folks recommend it. It seems like I buy everything else thru Amazon. I have purchased 4 unlocked GSM phones on eBay. No, none of the phones came with a Sim Card. (I purchased in the US from T-Mobile) 2 phone were Tri-Band; the 2 other were Quad Band. Of the 4 phones, 2 had absolutely no problems. 1 phone would not boot properly, and I returned it. One phone was not unlocked, and I had to have it unlocked in the FarEast. (The closest cellular dealer near my hotel charged me $30 US). Of the 2 phones that had issues, I got a refund from the sellers without requiring a protest to PayPal.
Sorry this was so long. Hope it helped someone. As someone would say YMMV (You're mileage/experience may vary)
Good luck....and plan ahead with this purchase. I believe in always keeping a GSM unlocked phone "around".