Consider Campi Flegrei, in the western suburbs. It's also volcanic and still bubbling. Try the Vulcano Solfatara in Pozzuoli.
The Amalfi Coast is easy to visit. Ferry from Naples to Sorrento, bus to Positano which makes a good base and then use the bus for exploring the coast (Fiordo di Furore is interesting).
The Greek temples at Paestum aren't too far away, and there's always Pompeii.
Around Naples itself, Spaccanapoli, or Vomero for good views over the bay and for the cloisters of San Martino.