Naples International Airport / Capodichino: Take the direct Curreri Viaggi airport bus to Sorrento station/Piazza Lauro: about 90 minutes, roughly €13, luggage included. It is the sensible public-transport choice; do not detour into central Naples to change trains unless the direct bus schedule does not work for your flight.
Napoli Centrale / Garibaldi: For comfort in the main season, take the Campania Express to Sorrento: around 50 minutes, reserved seating, air-conditioning and luggage space; roughly €15, generally mid-March to late October. On a budget, use the Circumvesuviana from the lower-level Garibaldi station—about 80 minutes and around €5. Board at Porta Nolana, one stop earlier, if you want a realistic chance of a seat; summer trains are crowded, dated and poor for luggage.
Naples Beverello ferry port: Use the hydrofoil to Sorrento’s Marina Piccola when sea conditions and sailing times suit: about 40 minutes and roughly €15–20 one way. It beats road traffic, but the port is below the centre; walk uphill or use the lift from the Villa Comunale area rather than assuming the ferry arrives beside your hotel.