I also recommend Poppaea's house at Oplontis, if you're not all Romaned out. Take the circumvesuviana train to Torre Annunziata. It's a single but very grand villa reputed to have belonged to the family of Nero's mistress. It has beautiful wall paintings, a swimming pool and the most fantastic 'wall' where you can see the layers of the deposits left by the eruption of Vesuvius in AD79 - really brings home how devastating it was. It takes about 45 mins to go round and is very evocative.
Be careful about Vesuvius - if you are short of time and it's at all cloudy, don't bother. If there's a cloud on the top you won't be able to see anything and it will be cold and horrible.
Pompeii and Herculaneum are both fantastic and the Archaeological Museum in Naples is wonderful, including the Alexander mosaic and the secret room full of Roman porn.
I love my ancient stuff. If you are a real obsessive like me, it's worth the trek (about two hours on a bus) to Paestum, where there are three of the finest Greek colonial temples in beautiful, if lonely, countryside.
I also recommend ice cream and seafood in Sorrento, limoncello and posh-deli shopping in Amalfi and an open-topped sports car along the Amalfi Coast road if you are feeling brave.