The best place to live in Rome is Trastevere. The city center is surrounded with endless boutiques, restaurants, and residential areas. Living in Trastevere is a perfect place to study abroad because it has everything from cafes and restaurants to bars and clubs. Also, everything is walkable in Trastevere, with famous tourist destinations like The Vatican museums and the Coliseum just on its outskirts.
You can use Roma Rentals to rent a cozy apartment located in the center of the city. Using non-university housing is definitely the way to go if your U.S. university allows it. Try to live on Viale di Trastevere, just walking distance to the Pantheon, Spanish Steps, Trevi Fountain, and all the restaurants/gelaterias!
Another nearby place to consider living is Testaccio. Located right next to Trastevere, Testaccio is known for its nightlife. With some great restaurants and clubs, many study abroad students live here as well.