Ischia
I personally find the speeding motorbikes and general chaos of Naples comforting, but for those who wish to escape, the island of Ischia is a natural refuge just a short ferry from the coast. I went in the beginning of June, and while the weekend was buzzing with mostly Italian tourists, Monday morning the island was quiet as they had all returned home to work.
Head down to Beverello Port, near the Castel Nuovo and the Municipio stop of the Napoli Metro. Make sure you know which port of Ischia you would like to go to - Ischia Porto, Ischia Forio, or Casamiccola (likewise on the way back, double check the port in Naples, as there is more than one). It takes only one hour with the fast boat, and the staff at the window is helpful to make sure you are on the right boat.
We opted to buy food at the supermarket and cook for ourselves, but we did have a great meal our first night at Ristorante La Baia del Clipper, the last of the restaurants lined along the port. We even sat at the last table, so we felt like we had the harbor to ourselves. With the relaxing sound of waves hitting the rocks beneath us also came mosquitos. Put on some bug spray before you go, and you will be able to enjoy a bottle of Biancolella, a local white wine, in peace.
Unmissable on Ischia is the Castello Aragonese. A 10 euro entry fee is worth having a few hours to walk around the grounds and gardens. We went around sunset, and with each turn we stopped and exclaimed at the beauty. There are even art exhibitions in some of the buildings. Clear space on your phone to accommodate all the photos you will want to take.
Mention you are going to Ischia, and someone will tell you to go the terme, or thermal baths. There are plenty to choose from, ranging free coves you can clamber into, to what look like swimming pools but are really the natural waters, and charge around 30 - 40 euro. We chose the Parco Castiglione, which is beautiful - it features pools of temperatures ranging from cold to 40 degrees celsius, and they even have a pier into the sea that I enjoyed a lovely solitary swim from. My group arrived later in the day, so we received a discount, but for the full day the cost of 28 euro is a steal. There is a cafe/restaurant there, but you can also bring your own snacks and water. They have towels and bathing caps for rent if you need for an extra 5.50€. For me, spending the late afternoon there until they closed was the perfect amount of time to relax in the hot water and take in the rustic beauty of the island.
We stayed in an AirBnB about a twenty minute walk from Ischia Porto. It was in a pedestrian area, the neighborhood between the port and the castle, with a huge terrace and a great bar/cafe across the street. We were able to walk everywhere, although we opted to take a taxi back from the spa as were so relaxed as to not be bothered to walk back. The taxis on Ischia have a standard rate depending on what area of the island you are going. You can find the chart online or in a tourist information office.
Ischia is definitely worthy of more time that I unfortunately didn't have to spend. I'm looking forward to returning one day to explore more of the island.