Rome is actually a large archeological park. Wherever one looks or goes digging, one finds remnants of classical antiquity. The most famous archeological areas are of course the Roman Forum, the center of ancient Rome and the world famous Coliseum. But one must not forget the Palatine, the Circus Maximus or the old Roman houses under the Church of the Santi Giovanni e Paolo al Celio. Outside of the city, you can visit for example Ostia Antica, the old port of Rome and Villa Adriana in Tivoli, the villa of Emperor Hadrian, both highly recommended!
Don't forget these are but examples of sites to visit. If you have specific wishes please contact me.