kt,
Your situation is very easy to resolve, but I would like to clarify one point.....
"We are Verizon customers, and we have heard that service is awful/non-existent in Italy"
I'm not sure why service would be "awful" since you'll be using European cell networks and not Verizon service. In my experience over many years, I have never had issues with service in Italy, especially when using a system that selects the strongest carrier in each area.
Are you planning to use Smartphones during your travels, or just basic voice + text phones?
If you don't want to use Verizon international roaming (assuming your phones have the necessary frequencies), then you have a couple of options.....
- Buy PAYG phones when you land in Italy - mobile phone shops are ubiquitous and you won't have any trouble finding one. If you don't see one right away, ask the staff at your hotel where the nearest one is. TIM is the largest network, but WIND or Vodaphone are also options. You'll probably have to present your Passports for purchase, and you'll need help setting it up as the menus will be in Italian. Be sure that you're clear on the method for topping-up when your minutes are depleted.
- Buy travel Phones prior to leaving home - there are a number of firms that provide inexpensive phones (or just the SIM card for those with unlocked phones). Have a look at the websites for Roam Simple, Cellular Abroad, Mobal, Telestial, EuroBuzz or Lebara. The advantage of this approach is that you'd have working phones as soon as you step off the plane, and wouldn't have to spend any time setting that up in Italy. Many of these use post-paid billing (calls charged to a credit card), so no issues with topping up. Phone numbers are often based in the U.K., but the travel phones operate everywhere in Europe with fairly consistent rates. I've found that basic voice + text service with these services is very cheap (depending on use), but if you add data the price goes up considerably.