To add to Angela's reply, you don't want a train "pass." You want individual tickets. Rome to Naples will be cheaper if you can buy it in advance on-line, and these will non-refundable and may be non-exchangeable. Or, if you don't mind the full ("base") price, you can buy them when you get to Italy.
Naples to Sorrento is on a separate train line, the Circumvesuviana, so you need a separate ticket. It's cheap and advance purchase is not possible, so just buy it in Naples at the station.
All the details of how to make this trip are in Rick Steves Italy and Rick Steves Rome.
It is not possible to leave Sorrento and make a plane from Rome back to the US that same day. Well, maybe if you leave at 4 or 5 AM, have a relatively late departure, and have hired a car for the whole trip Sorrento-FCO (costing several hundred euros). Otherwise, you will need to leave Sorrento the day before. The train from Sorrento to Naples is 70 minutes; the train from Naples to Rome is 70 minutes; and the train from Rome Termini Station to Rome Fiumicino Airport is about 35 minutes. And, of course, these trains are not synchronized, so you have to allow for time between trains. And, you want be at FCO three hours before a flight nonstop back to the US (and at least two hours before a flight connecting in Europe back to the US).