The Neapolitan Riviera is easily accessed by train (the Circumvesuviana line runs the length of the coast) or by SITA bus. If you're heading for the islands, you must first make your way to Naples' port (Molo Beverello) to take a ferry.
The integrated Unico Costiera travel card (www.campania.it) covers travel all over the Amalfi Coast (bus and train) but excluding Capri and Ischia (both islands have their own tickets). Tickets cost €2.40 for a 45-minute journey, €3.60 (90 minutes), €7.20 (24 hours) and €18 (3 days).
There are plenty of car hire options, though I don't recommend driving (for reasons set out here - Neapolitan Riviera car hire).
Most travellers heading for the Neapolitan Riviera will be flying in to Naples' Capodichino Airport. For information on getting to Naples and other destinations on the Neapolitan Riviera from the airport, visit Neapolitan Riviera flights, Amalfi Coast flights, Sorrento flights and Capri flights.
Once you arrive at your destination, getting around is remarkably easy. The SITA bus (www.sitabus.it) operates all the way along the Amalfi Coast and is, in my opinion, the easiest way of getting anywhere, whether it's to Sorrento, Amalfi, Positano, Ravello or back to Naples. Buses are frequent, running every 15 minutes at peak times and between 30 and 60 minutes at other times. For up-to-date times, check the website or you will find a schedule at each bus stop.
For ticket information and prices see the Unico Costiera website (www.unicocampania.it) but at the time of writing, a 90-minute ticket (to be stamped in the machine as soon as you board the bus) cost €3.60; a 24-hr ticket costs €7.20 while a three-day 3T Tourist Travel ticket (very useful, this one, if you're planning on moving around) costs €20.
In Positano, there is a small local bus that runs down Via Cristoforo Colombo to the main square and back up again costing €1.10 per journey. This is ideal for those with luggage but most others will enjoy the walk down (though perhaps not the walk back up so much).
No such bus is needed in Amalfi as it's all perfectly walkable.
Another, and much more fun, way of getting from place to place along the coast is to get the boat or ferry. This is easiest from Naples (Molo Beverello or Mergellina), Sorrento or, at the end of the Amalfi Coast, Salerno, as these have the largest ports and, while boats dock at Amalfi, Positano, and many other destinations along the coast, they won't run when the seas are rough (at the very start and end of the season). For details of times and prices, see the Metro del Mare website (www.metrodelmare.net) but, at the time of writing, fares were Amalfi–Positano €9 (20 minutes), Amalfi–Sorrento €11 (50 minutes), Amalfi–Naples €15 (100 minutes).
Accessible by bus, train or ferry, Sorrento is the easiest place to reach and to visit other destinations. Coming from Naples, the Circumvesuviana train (www.vesuviana.it) can be found on the lower level of the Central Station at Piazza Garibaldi and takes one hour. Once you're in Sorrento, there are three local buses that run variously down to Marina Grande (the smaller of the two ports), Marina Piccola (where the ferries come in) and out to Piano. All three stop at Piazza Tasso. Your Unico Campania tickets are valid in Sorrento. Ferries (see the Metro del Mare website) run more frequently from Sorrento – four times a day to Positano (€9, 40 minutes) and Amalfi (€11, 60 minutes), and six times a day to Naples (€6.50, 50 minutes).
As mentioned above, if you're travelling to Capri early or late in the season you'd be advised to travel from Naples or Sorrento as the boats from Amalfi and Positano are less certain to be running. Once you're on the island there is a bus service that covers the whole island but has its own ticketing (you can't use your Unico Campania tickets here). Unico Capri tickets cost €1.40 for a single journey. The funicular railway runs from the Marina Grande ferry port (where you arrive) up to the La Piazzetta (Piazza Umberto) and costs €1.40. From here you can catch a bus to Anacapri. The taxi service which runs from Piazza Umberto has a starting fare of €5.50 with each 70 metres costing an additional .20 cents and a minimum fare of €8. There are set prices to popular destinations. For details on Unico Capri, taxi fares and tariffs, and up to date ferry times and prices, see www.capritourism.com.
For those looking for a slightly more tailormade excursion around the island, I recommend contacting Capritime.