Well then you must be in Rome for the soccer game!
Your train connections will be very simple.
On arrival- what time do you arrive? There is a 1:53 train direct to Florence from FCO. That’s usually too long to hang around but might work for you.
Otherwise take the Leonardo Express to Roma Termini- very easy- just follow the signs at the airport.
From Roma Termini take a fast train to Florence.
It’s best not to book either in advance in case your flight is delayed, etc. If you download the app you can purchase your tix as soon as you land.
(It’s too late for any discounts now anyway)
From Florence direct fast train back to Rome.
From Rome to Naples or Sorrento- direct fast train to Naples.
If going to Sorrento you will then transfer to the Circumvesuviana- local train- go downstairs for that. There are directions for this here (do a search) and on the Sorrento Insider sight.
If you decide to stay in Naples- take direct train to Salerno- then bus, driver or taxi to Amalfi.
Sorrento books up early- you are only 2 weeks out so accommodations might be very limited there although it is not height of season yet.
Salerno is also a nice place to stay- not touristy at all, inexpensive lodgings, can get to Pompeii easily, it's got a loverly waterfront and some very good restaurants. The evening passiegiatta there was so delightful. Paestum (Greek temples) is amazing and an easy train ride day trip.
You will take the Circumvesuviana to Pompeii from either Naples or Sorrento. The stop you want is Pompeii Scavi
Many people prefer to hire a private driver to get them from Naples to Sorrento- that will cost about 120 euros but it is door to door and easier than Circumvesuviana with luggage.
We used this company and were very pleased. You may also want a driver to take you from Naples or Sorrento to your hotel in Amalfi, then from Amalfi to Salerno. The other option is bus. (No trains on the AC)
While in Amalfi don't miss visiting Ravello. Fabulous views from the Villa Cimbrone.
www.topexcurionsorrento.com
From Amalfi back to Rome- taxi, driver or bus to Salerno. Then fast direct train to Rome.
What time is your departure flight? We just stay in Rome on last night and have no issues but our departures are usually around 11 am.
Trenitalia- and get the easy to use app.
https://www.trenitalia.com/content/tcom/en.html
Be sure to set the clock to 00:00 so you see all the trains that day
Station names are:
Roma Termini
Firenze SM Novella
Napoli Centrale
Salerno
Circumvesuviana
https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule
Sorrento Insider- lots of practical info
https://www.sorrentoinsider.com/
You would have to visit Capri from Naples or Sorrento- not sure you have time along with Pompeii. You might consider doing the Capri day trip with Mondo Guides. It was the highlight of our time there- tons of fun. We also used them for our Pompeii tour.
https://www.sharedtours.com
From the Sorrento Insider site:
Ferry routes from Sorrento to Capri and Naples are active all year round. Fewer ferries run in the winter months and the hydrofoil to Naples only runs from Monday to Friday. Ferries to Positano, Amalfi, Salerno, Ischia, and Procida run from April through October and there are more ferries routes to Capri during these months. You need to arrive at the port about 15 minutes before departure to board your ferry.
Bus schedule
https://www.sorrentoinsider.com/en/sorrento-to-positano-bus-schedule
General train info:
https://www.seat61.com/train-travel-in-italy.htm
Buy your tickets on the official site- not a reseller like ItaliaRail. There are 2 train companies- Trenitalia and ItaloTreno. We just always book with Trenitlalia for simplicity sake, but check both and choose the best time/price for you.