On your first visit, you have to stop by all of the main tourist destinations and see double late at night on the Spanish Steps. Vatican, all that. It's required. Generally-speaking, I do not think much of Rome itself as a city to visit, but I feel the same way about Venice so ignore me if you want to.
However, the best thing to do in Rome is to take a tour of Ostia - Ostia Antica. It far surpasses Herculaneum and Pompeii as a preserved Roman urban site. It's the old port of Rome, the ancient harbor which silted up but was never destroyed. A stroll around there will make you realize that the Romans lived quite similarly to the way Americans live today.
Central heat, fast food, running water, sanitary toilets, shopping areas, multi-story apartment buildings, whorehouses, gambling dens, paved streets, warehouses, factories, etc. They did not wear togas, either - except the senators and their ilk. They did not have tomatoes.
A good intro to the "Old Port": Ostia Antica