You have lots of choices for a phone. Kiosks in airports, malls, departments stores and small stores all sell phones for different plans. I found at least three options on my last trip for a phone, charger, SIM and some minutes for 25 euro. Vodaphone and Orange were two of the companies; I can't recall the third off the top of my head. I'd say those three were the tip of the iceburg. If I'd looked hard, I could have found more choices.
The SIM is only "local" in the country where you purchase it. You are still roaming when you leave the country. The rates aren't as bad as most American plans but they can add up. You may want to buy a phone at your first stop, use up your minutes on the SIM, then get a new SIM when you get to Italy. It really depends on how much calling you anticipate doing.