Naples is served by various low-cost and other airlines from the UK, with the flight time from London two hours 45 minutes. You could also easily fly to Rome about a two-hour drive north of Naples and take the train (or drive) from there.
For trains from the UK to Rome or Naples, or for onwards trains and buses from Naples (including the narrow-gauge Circumvesuviana Railway to Sorrento, Pompeii and Herculaneum), see the ever-helpful Man in Seat 61 (seat61.com).
There are ferries from Naples to Capri and Ischia, and from Sorrento to Capri, plus cruise-boats for taking in the coast and islands.
Car-hire is useful if not essential, although you’re best off without one on the narrow windy roads of the Amalfi Coast, where parking is an issue too.