Flight
The airports we recommend are FCO in Rome or NAP in Naples (note that Naples is mostly connecting flights!) From either airport there are options for trains & busses, and a ferry from Naples
Train
From Rome There are high speed trains to get to Naples, at which point you can transfer to train/bus/ferry. The train from the airport connects directly to Roma Termini, where you can purchase a high speed ticket. (Note: there is a crazy surcharge if you board a train without a ticket for this leg of the journey. )
Train
From Naples The train to Sorrento is more similar to a metro. The train station it leaves from is beneath Naples Centrale (where the Rome train arrives). There is an elevator with signs to Circumvensuviana - which you can ride to the end.