Rome has a million things to do and see.
The imperial ruins at the Forum. A 'gotta see' place in Rome. Amazing Imperial Rome buildings and grounds remain. The imperial forum, Capitoline Hill and the Colosseum make up an unforgettable experience. If ruins are ever to be an interesting place to go, this is it.
Piazza Novana. The center of activity in the Roman city at night. Abuzz with people. The Pantheon and Trevi Fountain are close by. What a great area. Stroll among these streets and Piazza at night, you will love Rome.
The Baroque Churches. Do not bypass the Baroque churches, they are decked out to the hilt, even today they are clearly over the top -- the artistic expression of religious ecstasy is very strong. They pulled out all the stops to express their tenets and stories of faith.
The old city, 'Centro Storico'
The Vatican Museum. This has to one of the best museums in the world. The extent and quality of the pieces are amazing, each area you go through seems to open past centuries.
Campo de Fiori
Trestevere
Palatine Hill. The home of the Caesars when they were in Rome. Extensive ruins overlooking the Forum and Colloseum, the chariot race track and the Tiber River. Hard to see how anyone could live here without going bonkers. It would be like living between a hockey arena and a Nascar track. This is, however, the very apex of Rome, politically, romantically, historically, and in any history books, where Romulus and Remus were suckled by the she-wolf and went on to live. The basis of the western world can be discovered by living between a hockey game and a Nascar event. Go figure. It is a beautiful place, though, highly recommend going here in the morning, when the views are fresh and the morning fog is drifting through the buildings, the Colosseum, the towers and the hills of Rome.
It is a walking city, everything is close by.