Verizon phones and pre-paid SIM in Europe
Verizon and Sprint are the two U.S. major carriers whose phones are mostly incompatible with use in Europe because their phones use a technology called CDMA, while most of the world uses GSM. But if you're a Verizon user, it's not hopeless. Much of this is true for Sprint, too, but I am not as familiar with their policies.
If your Verizon phone is one of those that has both CDMA and GSM capability (Verizon calls them "Global-Ready" and if you've had your Verizon account for more than a few months, Verizon will give you a code to "unlock" the phone--that is, to defeat the software lock that keeps you from taking the phone to another carrier. They have that "loyalty lock" because they subsidize the cost of handsets when you sign up for a plan and make the subsidy back over the life of the plan.
With that unlocked phone, you only need to remove the SIM card that came with the phone, and replace it with the prepaid SIM from TIM or other vendor. There are also a couple of settings you need to change. If you go to the Tips, Tools and Deals/Phones forum here on Frommers and search Verizon and unlock, you should find a post I wrote a year or so ago on step-by-step how to do it.
If your Verizon phone is NOT "global-ready" you will need a GSM phone, and TIM will sell you one fairly cheaply--I believe under $40. The plus, of course, of using your own phone is that all your contacts and everything else are all there! If you find a GSM phone available cheaply in U.S., be sure that it is "quad-band" or at least that it has the 2 bands used in Europe: 900 and 1800 (US phones use 850 and 1900).
Rates for calls back to the U.S. will be significantly higher than in-country calls. One way around this is to use a "callback" service (see link below for a list). These work for you because incoming calls to your cell in Europe are generally free. You sign up with the service, and give them a credit card to charge. They give you a number to call (usually a U.S. number) but the number and where doesn't matter because what you do is dial it, let it ring once and hang up. The service recognizes your number as registered with it, and immediately calls you back. When you answer, you'll hear a U.S. dialtone and you'll dial your number. After you hang up, the service charges your card a few cents a minute.