Rome has a lot to offer, a never ending list of notable places, I would need a book to give you a proper list.
However, if you want to grab some ideas ad suggestions by a local:
The Roman National Museum. It hosts the most important collection of Roman sculptures, mosaics and paintings of the city, maybe even more important than the Vatican Museum collection but with the advantadge that it is never crowded (I do not really understand why…). If you go do not miss: the Boxer statue, the Portinari sarcophagus and the house of Livia paintings. With the same ticket you can also visit other 3 museums: The Domitian Thermal baths, Althemps Palace and the Crypta Balbi.
Another great museum to visit is the "Capitolini", that has also a wonderful rooftop cafeteria and and incredible view on the Forum (not to mention the amazing art collection). It is by the way the fist museum ever opened to the public in Europe!
The most astonishing ruins in Rome are no doubt the Thermal Baths of Caracalla. Huge pillars and vaults built tin the IIIrd cent., are to me the highest masterpiece of Roman engineering you can find
Have a walk if you can in Trastevere, it is a very nice district, full of charme, art and good food.
Do not miss the main church, Santa Maria in Trastevere, the pastry shop “Biscottificio Innocenti” in via della luce and the pizza and supplì street food style at “Venanzio, la Casa del supplì” in via San Francesco a Ripa.
Other nice discoveries are the undergrounds of San Crisogono church (ask the old man in the sacristy ,he will open you the door and turn on the light with a little tip) and the nuns choir of Santa Cecilia church (with 3 € you can go on the 1st floor and admire the wonderful frescoes by Pietro Cavallini)