From Rome Fiumicino Airport
From Fiumicino Airport take the Leonardo Express, train to the city's main Roma Termini railway station.
From here you can catch trains to Naples.
There are various types of train which run between Rome and Naples:
Eurostar Freccia Rossa (TAV): this is by far the most comfortable option. In little more than an hour, you will arrive at Naples' Piazza Garibaldi station.
Italo: They are mid-range trains and connects Rome to Naples in about two hours.
Intercity: these trains take about two hours to travel between Rome and Naples.
Interregionali: these trains stop at every station in just about every little town dotted along the line. Not surprisingly, the journey from Rome to Naples takes roughly 4 hours
Once you arrive at Naples Stazione Centrale -Piazza Garibaldi railway station, follow the direction signs to the Circumvesuviana and from here take the train to Sorrento.
You'll reach Sorrento in about an hour.
Outside Sorrento railway station you'll find autobus Sita headed to Positano-Amalfi or to Sant'Agata-Nerano.
From Rome Ciampino Airport
If you land at Rome's Ciampino airport, you can take a bus to Roma Termini railway station and from here follow the directions above.