You are good on the train (unless they are on strike) from Rome in both directions, it goes to Firenze and also to Salerno, which is pretty close to Positano. This is the high speed line (Pendulino) and it is a couple of hours from Rome to Firenze and about the same to Salerno, although I think you still have to connect in Napoli...
You can catch the Circumvesuviano train in a number of different communes just north of Positano. It's a slow train that will take you to Pompei and Herculaneum, or if you are so inclined, to Napoli. Pompei is a can't miss, it's awesome, Herculaneum as well.. A lot of the relics have been moved to the museum in Napoli, but just walking around the two dead cities is fantastic. I am not a fan of Napoli, I have been there twice and both times were kind of awful due to pickpockets, beggars and overall filth.
Firenze is worthy of several days. I would suggest to spend most of your time there. Head to the city center and just walk around. Check out the Galleria Uffizi, L'Accademia, the Medici Palace, the Boboli Gardens... Climb the stairs in the Duomo and check out the incredible artwork there. Don't wear shorts or have bare shoulders or you will not be let in. We have always stayed at the Hilton Metropole. Good location and a free shuttle to the city center.
Rome is good, but so crowded. If you go to the Vatican, be prepared to fight Chinese tourists tor space the entire time.
Lots of great restaurants, too many to mention. Take cash. Lots of Euros, as the credit card machine will probably not be working... Don't know why, but in Southern Italy, they never do.
Buon viaggieo!