Avoid midsummer if you can - in August you can barely find beach space to lay your towel. Come in spring (April-early June) or autumn (Sept-Oct) instead: it's quieter and cheaper, the weather is perfect for gentle exploration, and swimming is quite feasible until early November (though in April it's chilly).
Note: flight, boat, train and bus timetables change constantly, and airlines come and go, so please do not rely solely on this information for your travel planning. Check with relevant companies, or a flight search engine like Skyscanner, first.
Naples Airport serves the city and the Amalfi coast. It's located 12km outside the city centre.
From the UK:
British Airways flies from London Gatwick to Naples.
easyJet operates year-round flights from London Stansted and London Gatwick to Naples, plus seasonal services from Bristol, Liverpool and Edinburgh.
easyJet flies from Berlin, Paris Orly, Milan and Basel.
Air France flies from Paris CDG.
Lufthansa flies from Munich.
Transavia flies from Amsterdam.
Helvetic flies from Zurich.
Air Nostrum flies from Madrid.
Aer Lingus flies from Dublin.
SAS flies from Oslo.
Air Malta flies from Malta.
Aerosvit flies from Kiev.
You can also fly to another Italian airport and change onto an internal flight to Naples. Alitalia flies from Milan (both airports), Rome, Turin and Venice, and Meridiana flies from Cagliari, Milan, Olbia and Verona.
Getting to Capri:
You’ll need to take a bus or taxi from the airport to Naples port, where boats to Capri’s Marina Grande depart several times a day (ferries take around 90 minutes, the faster hydrofoils around 50 minutes). Capri is also served by ferries and hydrofoils from Sorrento - the crossing takes 25-40 minutes. It’s worth buying a ticket in advance during high season, when boats become very busy with day-trippers.
If you are just visiting the Amalfi coast (and Naples), don't hire a car: the roads, though spectacular, are clogged with coaches (traffic police operate one-way systems through the narrow villages) and demand extreme attention. Parking is expensive (over €30/day in Amalfi and Positano), and the SITA buses and Metro del Mare boats are frequent and reliable. In addition, there are Circumvesuviana trains betwen Naples, Sorrento and Pompeii.
Capri also has an efficient bus network, making travel between its towns quick and easy. Note that non-resident cars are not allowed on the island.
If you are exploring a wider area however, see our car rental recommendations.