[QUOTE=Desert Nomad]
A coupe of definitions… GSM is a communication standard (think of it like AM and FM radio). A SIM card is a small (roughly 1/2" x 3/4") plastic card with a bit of circuitry on it. They are inserted into a GSM phone and are what gives it a phone number and connection to the network.
Your best best is to buy an unlocked 4 band GSM phone (GSM is the standard everywhere except the US and Japan). There is GSM service in the US from AT&T and T-Mobile, but Verizon uses a different technology and phones that work with Verizon will not work on a GSM system.
I have a Sony 520i which is a 4 band GSM phone. It cost me about $200 in Dubai. Buying unlocked phones in the US will be more difficult, but I am sure there are online places where you can get them. Being unlocked means you can insert any company’s SIM card in any country. A locked phone will only work with a specific company’s SIM card.
The 4 bands are 850Mhz, 900, 1800 and 1900. (Going from memory here). More or less, most of the world uses 900/1800, the US uses 1900 and the 850 band is the newest and is used in a few places. Thus a 4 band phone will work on any GSM system worldwide.
When you arrive in a country, you can buy a local SIM card that you insert into the phone. This gives you a local phone number and usually a bit of credit… sometimes as much as the SIM card costs. So for €20 you get a SIM card with perhaps €10 of calling credit. If you use up all that, you can buy scratch off cards with a code to add more credit.
Outside the US you will never pay to receive calls… that is a weird and uniquely US thing.
Having a local phone makes it much nicer to travel. A couple months ago my wife and I were in Tunisia. The first morning we bought a Tunisian SIM card ($5), added credit ($10) and for two weeks we had a phone. Being able to make a hotel booking from a train is much better than finding a phone office and using coins. 
Note that many pre-paid systems will not let you roam so a German SIM may not work in Italy. This is not always the case tho. A Czech SIM works almost everywhere (even Thailand).
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That’s one approach. I need to have a cell phone number that is the same no matter where I am in the world (for work reasons), so your method won’t work for me. I went with a standard GSM phone (T-mobile). Although calls are expensive outside the U.S., it does work in most countries.
Ed