Surroundings
POZZUOLI
Visit the Solfatara. A camping is right there. The train stop (metro linea 2, Pozzuoli Solfatara) is about 1km from the site and right next to the Anfiteatro Flavio. If you go down towards the coast, the pubs there are quite popular.
Rione Terra
SORRENTO
A website with some hiking trails (in italian): http://www.sorrentohiking.com/. I have only been in Ieranto.
ISLANDS
I can strongly recommend both Procida and Ischia. The advantage of Procida is that it’s very close (about 35’ from Pozzuoli), and quite small and pleasant. So you can actually visit most of it on foot. Ischia is really beautiful (ask Frau Merkel), and I would really like to hike Monte Epomeo one day, see here. Also Ischia is easily reached from Pozzuoli. I would consider spending one night there. And then Capri, no need for an introduction.
CAMPI FLEGREI
The whole area of Pozzuoli and surrounding is called Campi Flegrei (burning fields). If you dive, you might visit the underwater archaeological park of Baiae http://www.centrosubcampiflegrei.it/. Also on my personal todo list. There is a nice beach where I try to go when I can, just underneath the Castello di Baia. Basically you have to go to this parkin lot, and from there a boat will take you to the beach. If you are looking for a nice restaurant/bar in the area, go to Batis, in Bacoli. The guy at the bar should speak a good english.
VESUVIO
You can drive almost to the top and then hike the last part.
POMPEI
Interesting to know is that an ideal completion of the visit is the Museo Archeologico Nazionale in Napoli, where many findings are displayed.