Starting with Rome to Naples by train, then you can proceed to Sorrento by private car service (door-to-door), or another, perhaps crowded train to the train station in Sorrento and then need a taxi.

If you're staying overnight in Rome, you could program the train connection at Napoli Centrale to connect to the upscale Campania Express train, thereby experiencing a much nicer ride on to Sorrento and avoiding the usually very crowded regular Circumvesuviana commuter train. You could also pre-purchase tickets for the fast train from Roma Termini to Napoli Centrale to get the lowest possible fare for your travel date. You have the choice of either Trenitalia or Italo Treno.

http://www.trenitalia.com

http://www.italotreno.it

https://www.eavsrl.it/web/sites/default/files/ORARI%20NAPOLI%20SORRENTO_dal%202%20Ottobre%202023_0.pdf

(The Campania Express trains are show in RED. The one-way fare is 15 Euros per adult. Regular fare on the Circumvesuviana train is 4.60 Euros but is not recommended if you have large or even medium-sized luggage.) You leave from Napoli P. Garibaldi, which is one level below ground at Napoli Centrale station. Very easy to get there by following the overhead signs. Allow for a minimum 30-minute connection, in case of a delay of the incoming train. It takes less than 10 minutes to actually connect between trains. Most tourists would leave Rome to connect to the 11:26 a.m. Campania Express train to Sorrento. That means departing from Rome around 9:30 a.m. You can purchase the tickets for the Campania Express online (you get a reserved seat, as on the fast train to Naples), or wait until you arrive at Napoli Centrale and buy them from the staffed ticket windows at Napoli P. Garibaldi (one level below ground). The Campania Express trains are usually not sold out, as locals won't pay the surcharge and take the regular crowded commuter train. Trenitalia (but not Italo Treno) get a small discount on the Campania Express.

The above becomes MORE DIFFICULT to program and RISKIER if you're going to Sorrento straight from FCO airport after landing there. (Riskier because you'll never know exactly when your plane lands, nor how long it will take to clear passport control and retrieve your checked luggage). The risk is forfeiting the train ticket purchased online in advance, especially if at a discounted fare.

Easiest, to me, would mean without changing modes so I'd lean toward the through bus. It will take longer, but you stow your bags once and relax until you're in Sorrento.

I hope this is the right link for the bus:

https://www.sorrentoinsider.com/en/rome-to-sorrento-bus-schedule

Yes, that's one of them. There are two that I'm aware of:

https://www.marozzivt.it/

https://www.flixbus.com/