Take the Circumvesuviana line from the station at Piazza Garibaldi (this requires different tickets from the giranapoli ones you use for buses and metro, so ask at a window - costs about 8 euros, I think). The trip alone would be worth it, as the landscape changes from incredibly built-up and densely populated on the outskirts of the city, through countryside with stunning views of Vesuvius on one side, and out over the Bay of Naples on the other side.

It takes about 40 minutes to Pompeii, which I'm not going to "do" in detail here - just go and see it. If you're a bit lazy, like me, there's far too much to take in, but even when you've given up reading the guidebook, it's a beautiful place to wander around, and because it's such a big area, it's still surprisingly peaceful even with lots of tourists there. Be sure to pick up the free mini-guidebook and map at the entrance.

Another half-hour or so on takes you to Sorrento, which is all picture-postcard pretty and small enough to wander around in a couple of hours at most. Lots of nice places to eat and reasonably good-quality "souvenir"-type shopping.

If you just want somewhere to have a coffee or snack with a great view over the bay, go down the Via de Maio from Piazza San Antonino and there are signs pointing you to turn right into something like the "foreigners' club" (can't remember the exact name!) - walk through the building and out to the garden, and you'll find a great view and some peace and quiet.