Mark and Kristy,
As the others have mentioned, your AT&T IPhone will work fine in Europe however a "normal" Verizon phone will NOT (unless it's one of the "international" models mentioned by Tim).
If your IPhone is "locked" you only have one option for travelling in Europe - roaming with AT&T. This can be a reasonable solution if you're only planning to use it for very occasional voice calls and texting. Arranging an international roaming plan with AT&T will reduce the costs somewhat.
However, it's NOT ADVISABLE to use the data portion of the IPhone, as it's gets to be VERY EXPENSIVE! Prior to arrival in Europe, it's a good idea to disable the data portion and E-mail as the phone will check periodically, which you'll be charged for.
"Does AT&T get good coverage in Italy?" When roaming in Italy, you won't be accessing the AT&T network per se, but rather one of the Italian networks (and the coverage is usually excellent). As you travel around Italy, you'll probably find that the phone chooses different networks, depending on signal strength. You'll receive an itemized bill from AT&T when you return home.
If your IPhone is "unlocked", you'll have the option of using either PAYG SIM's with an Italian network or a SIM with one of the "travel phone" networks such as Call In Europe, Roam Simple, Cellular Abroad, Mobal or Telestial.
If the Verizon phone is not an "international model", two choices to consider: purchase an unlocked GSM quad-band phone on E-Bay or purchase or rent one from one of the firms mentioned above.
Happy travels!