Lucca to Sorrento

Look at http://www.rome2rio.com/s/Lucca/Sorrento - looks like train is the best option, least time consuming and least cost. I like to use this website just to see what the available options are; just be aware that the prices they quote are approximate and often not the cheapest you can get and also that their time estimates, especially for driving, are usually understated.

Train from Lucca to Naples (Napoli Centrale). You will need to change trains at least once in Florence (in some cases, in Pisa as well).
For schedules and prices see:
www.trenitalia.com

Then once at Napoli Centrale, walk to the Circumvesuviana commuter trains (it's at the lower level), but the ticket and get aboard the train going to Sorrento.