If you have lots of PC points, forget the HI and stay at the IC De La Villa in Rome. It's a fabulous hotel at the top of the Spanish Steps. Check out http://www.ichotelsgroup.com/h/d/ic/1/en/hd/romha?irs=y

Most of the other tips are good. The weather in June is usually great. Not too hot. I live in France, very close to Italy and I drive over often. Driving in large cities is a challenge, but one that I've managed to conquer several times in places such as Milano, Torino and Firenza. And you do need a car to visit the outer regions.

Someone said something about car rental insurance for Italy? I have never had to do that. Perhaps I have a special Amex agreement or its through my employer, I'm not sure. But all you have to do is check before renting.