S.W.
I lived in Italy for 5 months a few years ago. I was based in Rome and traveled all over Italy and Europe - even though I lived and studied there, I still didn't see everything I wanted to see.
The trains are a great way to get between cities. I wouldn't rent a car - the driving rules and traffic can be very scary, not to mention the high price of gas. Italy is practically designed for tourists. lots of things are within walking distance (take comfy shoes). If you plan on visiting cathedrals, take long pants/skirt with you and shirts/blouses with sleeves - some places are very strict about dress codes.
Small family run hotels and B&Bs are usually your best bet - inexpensive and very friendly (remember, these are people who depend on tourists like you for their livelihood). The one Lisa recommended is in a great part of Rome - Trastevere. Just a short walk to the metro, which takes you practically anywhere you'd want to see in Rome.
Look into guide books by Rick Steves (or even DVDs). He's great about reviewing quaint, not-so-touristy locations, including hotels and restaurants. Frommers or Let's Go are also good guidebooks.