Can I get a train or bus at the Rome airport that will take me directly to Sorrento? I'd like to get a direct flight to Rome from NYC, rather than a 1 stop layover and go to Naples. The plan I'm trying to work out is spending the front end of our (wife, 15 yr.old and I) vacation in Sorrento, using it as our base camp for 3 days; then taking a bus/train(?) back to Rome before embarking on a cruise.
You can take the Leonardo Express non-stop from FCO to Termini Station in Rome. Then take a train to Naples and get a private driver or go to Naples airport to get a dedicated bus to Sorrento. There is also a commuter train from Naples to Sorrento, Circumvesuviana, but it like a subway car, crowded. I wouldn't take it with luggage. There are many private drivers; I think we paid around €120 from Sorrento to Naples airport a year ago.
Since you are ending with a cruise I suspect you won't be traveling light as far as luggage. I would take a connecting flight to Naples from Rome FCO and then the bus to Sorrento, taxi to your hotel there.
this site : https://www.rome2rio.com/s/Rome-Airport-FCO/Sorrento shows a bus from FCO to Naples. You then need to take some other form of transport ( boat, rail, bus, or "taxi") to get to Sorrento