Any GSM phone should work in Europe, but you should try hard to get a 'dual band' GSM phone which supports both 900 and 1800 bands. A single band phone will work but will be seriously limited. Similarly, you should try and get an 'unlocked' phone - a 'locked' phone will recognise only its original vendor's SIM card (meaning you can't change SIM card at a later date). Note that Nokia phones can easily be unlocked for free, or for a token fee.
There are ways to rent a phone, and your US provider may be able to advise if you want to use your US number. Many providers, however, charge enormous 'roaming' fees for making and receiving calls.
Another option is buying a 2nd hand GSM phone, and buying a pre-paid SIM card for the network of your choice in the country of your choice. You will get a local number, and the ability to make cheap local calls. These are easily available in high-street shops and even supermarkets, although you may need to provide a local address in order to activate the SIM card.
If you want to call back to the US, then there are a number of 'dial-through' services which can give you international calls for the price of a local call.