On a three week trip to Italy we are spending a week in Naples and are interested in good day trips, rail, bus, or ferry. I thank you in advance.
Pompeii is a given but I wouldn't rule out Sorrento or the Amalfi Coast.
Baia, Pozzuoli, Caserta, Cumae, Benevento, Paestum, Sorrento
Pozzuoli, about 45 minutes from Naples. Train.
Flavio Amphitheater, Rione Terre.
Reggie Caserta. 40 minutes. Train.
Ischia, Prodica, Capri. All by ferry.
Paestum. 38 minutes. Train.
Castel Sant’Elmo for the best views over Naples.
Take the funicular up.
Agree with Susan, Baja has a wonderful underwater archeological park that you can do by glass bottom boat.
Cumae the Sibyl cave was closed the last time we were there, but the other ruins were a nice experience.
Better to stay in Sorrento and day trip from there.
Go to Pompeii, Capri, The Amalfi Coast.
I highly recommend Ischia if you are looking to go to an island for the day.
The Castle is stunning and you could spend a few hours there.
We went to Procida last trip, and it wasn't nearly as nice.
Herculaneum (Ercolano on the circumvesuviano). I have never been able to decide which I like more - Herculaneum or Pompeii (they are very different - Pompeii is larger with the major buildings, but somehow I feel a far greater sense of the disaster in the former).
Beyond Paestum, and a fair walk from the station, Agropoli has a decent castle. I'd recommend Castellabate, but it is tricky to get to without a car (There are buses from Agropoli, but they are infrequent and stop at "Santa Maria di Castellabate" from where you get the local bus up the hill.)
Benevento has a great Roman Triumpful arch, and the remnants of an amphitheatre from the same period.