Peter,
Are you planning to carry on with business when you're travelling and receive calls in Europe? That will have a bearing on which phone option to choose. Be sure to consider the time difference for making & receiving calls.
As you noted, your regular Verizon phone won't operate in Europe as it uses a different technology (Verizon CDMA vs. GSM which is used in 85% of the rest of the world). One unique characteristic of GSM phones is that these use a SIM card which is a small chip about the size of a postage stamp (normally sits under the battery). This contains subscriber information and the phone number.
As a previous post mentioned, a "locked" phone will operate only on the network that originally sold it, while an "unlocked" model will operate on any GSM network simply by inserting a new SIM card.
One other important point is that to travel in Europe, the phone handset must also be a quad-band model. Europe uses 900/1800 MHz frequency bands, while North American GSM networks use 850/1900 MHz (there are additional bands for data, but you can ignore those for now).
A few possibilities.....
You could rent or buy a phone from one of the "travel phone" firms such as Call In Europe, Roam Simple, Cellular Abroad, Mobal or Telestial. They also offer SIM's for those that have their own phone. The advantage of using a "travel SIM" is that the rates would be the same in all the countries you'd be travelling. Be sure to check their rates carefully so you have some idea on what the costs might be.
You could buy an unlocked phone off E-Bay or elsewhere, and buy SIM's in Europe when you land. However, keep in mind that if you buy a SIM in Italy, when you use this in other countries you'll be "roaming" and the rates will be higher. I don't feel that buying a new SIM for each country is a practical solution, as the number will change each time.
Good luck!