I just spent 8 days in the Campi Flegrei area in the very small town of Lucrino, close to Pozzuoli. I had a great time with my 16 year old daughter. We got around mainly by train. This is cheap(1 euro per 90 minutes), easy and reasonably comfortable. Validate your ticket in the little white box at the stations.
Latest trains from Naples were at 21.20 and that suited us fine, as we didn't feel like hanging out in Naples too late after dark. We had to walk a few back alleys to the Cumana station, but we never, ever felt uneasy. We talked to people a lot, asking for directions etc., and everybody were ever so sweet and helpful. Very few speak english, so I'm going to learn much more Italian before my next visit. But still: a few words really go a long way!
We loved Naples. It's loud, exhausting and rough around the edges, but it's fantastic. We took the neccessary precautions and paid attention to our gut feeling and we didn't see or experience anything disturbing.
Buy food from the heated counters along the streets in the evening, almost everything tastes delicious!
If you should decide to go west, Pozzuoli is a very nice town. They actually have the third biggest amphi theatre in Italy. It measures only a few metres less across than the Colosseum in Rome!! On the outskirts you even find a bubbling and spitting volcano-crater. The town itself is very pleasant and we actually did most of our shopping there. Great coffee and ice cream too. Ferries to Ischia and Procida leave several times a day from the easy accessible harbour.
We ate most of our evening meals where we stayed,- in Lucrino. The reataurants "Il Capriccio" and the "Lucrino" both had the most delicious seafood, pasta and pizza. To our taste the pizza was actually better here than in Naples!
The beaches stretch out along the bay, but they were a little disappointing. We managed to find a spot or two which were acceptable and the water itself seemed to be clean enough, but the surroundings were not too inviting. The countryside on the other hand is beautiful, green and lush. Take a walk to lago Averno, in ancient times believed to be the entrance to Hades. Close by is Monte Nuovo,-'Europes youngest mountain only 500 yrs old. There is a lot of history here, and well worth a trip.