Naples to Positano

Depending on where you are staying in Positano, take the train to Sorrento and the bus, which is directly outside the train station, to Positano. This works well especially if you don't have a lot of luggage. Be sure to ask the driver to tell you where to get off. In any event, take the train to Sorrento and, if you are hesitant to take the bus, have a private car take you to Positano. Rick mentions a couple of private car companies in his book. You'll find the bus is the best way to get around the Amalfi coast and the locals use it so we always enjoyed the chatter of Italian all around us on these buses.

Sometimes it's worthwhile to spend the extra money, I think. I am understanding that you will be arriving at Naples airport after a long flight, maybe connecting flights, likely tired, maybe jet lagged, certainly excited to start your visit. If you google "taxi Naples to Positano" or something similar, you will find that a private car from the airport to your hotel in Positano will take you and your bags door-to-door for just about 100 euros in about 90 minutes. I'd do that. Certainly you could get to a train or bus to Sorrento, maybe crowded, wait for it to leave, then find another bus in Sorrento, also maybe crowded, wait for it to leave, arrive in Positano and then figure out how to get to your hotel, probably by taxi or on foot with your bags. I myself would choose to spend the relatively few extra euros, enjoy the scenic drive, and get on with my visit to the wonderful Amalfi Coast in the quickest and most relaxing way possible.

or an alternative between the 2, take a train to Salerno (it would involve taking the 3 euro airport bus to the train station) from Naples, then walk 300 yards to get to the tourist port to take a ferry.....

Naples-Sorrento
http://www.curreriviaggi.it/inglese/E_index.html
located outside the arrivals hall Capo airport, 10 euro, purchase from driver, 1H15m

Sorrento-Positano Taxi
http://www.taxiautofare.com/it/taxi-fare/Naples-taxi-fare-from-Sorrento-to-Positano
last year 17 euro, 30 min.

Sorrento-Positano Sitabus
http://www.theamalficoast.net/orari_sita_bus_timetables.html
1,60 euro, around 45min. Daily ticket 4,50 euro

If you are visiting Pompeii and any other sites both on the AC and in Naples, look at
http://www.campaniartecard.it/ Tutta la Regione
3 day 32 euro includes transportation

We go to Sorrento on holiday 2/3 times a year. IME the cheapest car service from Sorrento to Positano is about 50 euro for 2/3 passengers. The local taxi services in Sorrento will be much more expensive. Eg it has cost us 16 euro to go from Sorrento train station to our lodgings overlooking Marina Grande, about a 20 minute walk but we had luggage.