I am planning to fly into Naples and then spend three nights in Sorrento and three nights in Positano. What is the most convenient way to get from Naples to Sorrento and then Sorrento to Amalfi? Thank you.

The most convenient (but also, of course, the most expensive) is to hire a taxi or a car service to take you. You can search "car service Amalfi" and get lots of suggestions.
Otherwise I'd take the direct bus from the airport to the port in Naples and take the boat across to Sorrento. I'd also opt for a boat from Sorrento to Amalfi rather than the busses which often involve pushing and shoving to get on, or else waiting for the next one a couple of hours later.

Hi dana,

If you don't want to hire a car, there is also bus service from Naples airport to Sorrento.

http://www.curreriviaggi.it/

Method 1:
Take the Bus from right outside Naples airport straight to Sorrento: http://www.curreriviaggi.it
Note: The last bus is at 7:00 PM, so it all depends on when you are arriving into Naples.

Method 2: Take the Airport Bus - ALIBUS - 3 Euros (for more info: http://www.gesac.it/en/alibus.html) to the Port. From there you can take the Ferry to Sorrento. Once again - Check the schedule to ensure you don't miss the last ferry.

Method 3 - Take the Alibus to central train station in Piazza Garibaldi. Take the Circumvesuviana commuter train to Sorrento - http://www.vesuviana.it/infoeng.htm

Method 4 - Private Limo service - write to them and see how much they charge
http://www.sorrentocarhire.com/
http://www.sorrentolimo.com/
http://www.sorrentocardrivers.com/02.asp
http://www.positanocarhire.com/?lng=en
http://www.amalfilimo.com/
http://www.seahorsecarservice.com/en/profilo.htm

hawaiihoneymoon: Thanks for the summarized transportation options. Please note that "sorrentolimo.com" and "amalfilimo.com" are one in the same company. People don't need to write them twice for information.