If you've ever heard of the saying "All roads lead to Rome," then that should give you an indication that there is literally so much to do here. Obviously it is a huge tourist attraction, so you can't miss the Colloseum, the Trevi Fountain or the Pantheon. Other ones that are equally worth noting are the Spanish Steps, Piazza Navona, and Campo de Fiori. The Vatican and Vatican city are absolutely incredible as well.
The neighborhoods in Rome are all different, which makes them great places to loose yourself in. I lived in the neighborhood Trastevere, which is an absolutely incredible neighborhood. Incredible food, people, and it is one of the oldest neighborhoods in Rome. There are also great American bars in this neighborhood. Prati is another great neighborhood and so is Monti. Loose yourself in these neighborhoods. Go into shops, try new foods from the delis, and you will not be disappointed.
Italians can be very tough to deal with. A lot of them don't speak a lot of English, and get frustrated with Americans super easily. Just smile and always be polite, and they will get over their judgments quicker.
The taxis try to rip of Americans and will start yelling you in Italian and tell you that they don't understand you. Stand your ground on the flat rate on the side of their door. They try to see if you will break and when you don't, they will realize and oblige to your requests.
Italy is one of the only countries that pretty much sticks to their kind of food. They don't like to do other food, because honestly they do their food so well. Go for the multiple dishes. Italians like to have their pasta as the first course, and meat for the second. Go for the lamb, it's one of my personal favorites.
The best thing you can do for yourself is to walk into a restaurant, and have them order the food and wine for you. We do this pretty much every time that we go out to eat, and we have never been disappointed. Try what they love and you will love it as well.