Yes, the phone will work fine in Italy with Verizon's international roaming service, if you pay for it. (Your phone already has a Verizon SIM card in it.) Verizon's $70/month plan seems very limited to me - only 500MB of data. I use maps on my phone constantly when I travel and easily use 100MB a day, so 500MB wouldn't last me long. The $10/day plan would be a better deal in my view if your trip is short - 7 days or less - and you don't get charged on the days you use it.
You can buy a TIM or Vodafone SIM card in Rome (go to a TIM or Vodafone store, like you would go to a Verizon store at home) to replace your Verizon SIM card while you are traveling, to get a lot more data for a lot less money - especially if you won't need to call/text home with your Verizon number while you travel. (Be careful not to lose your tiny Verizon SIM card! Maybe bring a tiny plastic case to put it in so you can't lose it.) Bring a paperclip with you to pop out the SIM card - the agent at the TIM or Vodafone store can do it for you, but you'll want to be able to swap SIM cards back again after you leave.
If your iPhone is brand new and purchased from Verizon, it MIGHT be locked - but after 60 days, Verizon claims they automatically unlock their new phones...