Barbara, except for major holidays, you don't need to worry about buying advance tickets. The Rome, Florene, Venice train runs are the heaviest travelled in Italy. Buy the tickets over there when you arrive.
55 trains a day run from Rome to Florence each day. Another 55 run the opposite direction. Most all are high-speed non-stop Eurostars. A few less make the Florence to Venice run. You won't be able to buy advance train tickets from Naples to Sorrento as the Circumvesuviana train system does not have reservations. And, you'll have to take a bus from Sorrento to Positano. No train there. Plenty of luggage space in overhead racks on the trains.
If you have an early flight out of Rome (before 8:30am), it is so wise to spend you last evening near the airport if you can. If your flight is after 8:30am, you could stay in Rome but near the train station so that you can take an early Leonardo express out to the airport.