Siena to Sorrento by train

Train to Rome. Rome to Naples
Switch to Circumvesuviana train or get a direct shuttle bus from Naples airport, a better option as the local train can be very crowded, more like a subway car, lots of grafitti, no room for luggage.

A better option would be the Campania Express that departs from the same underground tracks used by the Circumvesuviana but has only reserved seats. Why take a bus from Napoli Centrale to Naples airport to transfer to a bus to Sorrento if there is a not crowded, direct train?

Another option would be bus from Siena to Roma Tiburtina station, a direct train to Salerno and then a bus to Sorrento.

I believe the (interminable ...) public bus from Salerno to Sorrento would require a change in Amalfi. Month of year makes a difference in this high tourism area.

Time of year? Campagna Express only runs during tourist season (and only about four times a day when it does run) -- but it sounds like a great idea if the times work for you.