Judging from the comments so far, people will probably disagree with me. However, I lived in Italy for a few years and would say:
1) Avoid Naples. It is a pit. Everything thing around Naples is amazing, but Naples is not nice. The awesome things around Naples include the Amalfi Coast, Pompeii, Sorrento, Capri, Pozzuoli Amphitheater, Ercolano (Herculaneum), Mt. Vesuvius, and much more.
2) I am not a fan of Florence from a tourism perspective. I mean, it is fine from a city perspective. However, most of it is a large modern(ish) city with a big Church (Duomo) and a fair amount of museums. So - if you want to see *another* church or are a huge fan of museums then head there. But, don't expect to walk around the city thinking "wow, wow, wow" like you will in Rome.
3) Day trip from Rome: Hadrian's Villa is quite nice. I know you can ride a bus or book a tour to get there. I had my own car and it was a short trip.
4) Day trip from Rome: Ostia Antica is not too far from the Rome airport and you can get there on the metro for cheap. Kinda like Pompeii but close to rome. Really liked it.
5) In Rome: MUST visit the "Scavi" / Necropolis. These are the excavations under St. Peter's and just completely amazing. Check it out
here. You *have* to make a reservation in advance. Make them as soon as you finalize your travel plans.
6) Colosseum: As someone else said, the lines can be very long. Buy you tickets in advance and you will skip right past the line!
7) Vatican Museum. Okay...now, I am not the biggest fan of museums. For example, I was sorely disappointed by the Louvre. However, I love the Vatican Museum. Again, lines can and often are extremely long so buy tickets in advance and skip the queue. Pro tip: the Sistine Chapel is part of the Vatican Museum. Go there last and there is a door on one end that is an exit and marked "tour groups only" or something like that. Just go through the door and it dumps you into St. Peters. If you wanted to go the "approved" method you would have to walk all the way out of the museum, back around, stand in another long line etc. It is at least a mile to do that. Just go through the door! (When in Rome!)
Hope that helps