I am going to Rome in a few more days and will be staying there for 5 days, with trips to Amalfi coast and possibly Florence. What sights should I reserve for from the US before I go so that I can avoid long lines? Vatican museum, for example?
Generally the advice is as follows:
Colosseum/Forum/Palentine (one ticket) - No reservation necessary. However, you'll want to buy the ticket somewhere other than the Colosseum, usually the entrance to Palentine Hill has little/no line. Alternatively, you can purchase the Roma Pass elsewhere, such as at Termini train station.
Borghese Gallery - reservations required. You can reserve online (must purchase ticket), by phone, or ask your hotel to do it. If you want to use the Roma Pass, you can't do it online.
Vatican Museum - the lines are not as scary as they have been. If you choose your time wisely, you will probably not have much in the way of lines. Alternatively, you can purchase tickets with a reserved time from the Vatican's own website. Or sign up for a tour.
You likely won't need reservations for any other sight in Rome. Should you choose to go to Florence, you will probably take the fast Eurostar Italia, which requires reservations, which you probably will want to buy directly at Termini Station when you arrive.
For Florence, you will want reservations to the Accademia and Uffizi Gallery. Again, your hotel can probably help or it's easy to call. The directions are in Rick's book (and posted here fairly frequently, just do a search).
Can't help with Amalfi Coast, but I suspect there won't be anything you need a reservation for.