Your time in Rome is precious, so do whatever you can to avoid the lines. It's been mentioned, but the lines at the Vatican and the Colosseum are very long. Purchase your Vatican Museum ticket with entry time before you go. Also for the Borghese Gallery. The Underground tour at the Colosseum lets you skip the line. That ticket also includes the Forum and the Palatine Hill. There's actually two lines at the Colosseum, the first one is just to buy the ticket. So if you don't do the tour, get your ticket at the Forum or Palatine Hill because it includes the Colosseum, and there's hardly ever a line.
Last year, they weren't selling the Underground Tour tickets online (maybe that's changed), so we had to get those over the phone. It was kind of a pain in the ass, and the operator was rather rude to my wife. But skipping those lines? Worth it!
When exiting the Sistine Chapel, go through the door in the back to the RIGHT. It says tours only, but I've never been stopped. By going out that door, you can get to St. Peter's directly, and without going through security again. If you go out the left door, then you have to walk all the way back to the start of the museum, and then around back to St. Peter's, and then through security to get in.
Make sure you're early for your entry time at the Borghese. Depending on the season, they'll resell your ticket if you're late. Also, there's an annoying bag check that you may have to use. The Bernini's are amazeballs, and totally worth the visit!
When I was in Rome last fall, the Trevi was covered in scaffolding. The time before that, half of the Pantheon was covered. It seems like St. Mark's in Venice is perpetually half covered in scaffolding. In Florence, the Duomo museum was closed for renovations. It's Italy; it's old. You just have to go with it. :)
Definitely walk around Rome as much as you can. So many amazing things to see and do. Relax in the piazzas and people watch. Take your time, and don't try to do everything on this trip.
Positano and Capri are lovely, but I'm not sure that I would take the time to go there (I've never been to the town of Amalfi). When I've gone to Capri, it was from Sorrento that we took the ferry from both times. If you do go to Capri, take the boat tour around the island for some spectacular views (it's inexpensive too). I would skip the Blue Grotto. Villa San Michele in Anacapri is worth the visit though.