There is fast express train service to Rome and points north, as well as points south. Naples is the ultimate terminus for the FR7 line of the Rome commuter rail network, which runs from Roma Termini to Minturno-Scauri, Sessa Aurunca-Rocca or Naples. There are also local Italian Railway trains to Pompeii, but for such short distances, it is easiest to take the Circumvesuviana commuter train.
caserta royal palace
Reggia di Caserta - A huge 18C royal palace and hunting lodge designed by architect Vanvitelli and rivaling Versailles. Magnificent, enormous park with lakes, rivers, statues, fountains and marvellous views.By train from Naples to Caserta about 40 minutes journey.
gulf of naples
Ischia
Capri
Procida
Sorrento
Positano
Amalfi
Pozzuoli
Phlegraean Fields
Tour the excavations of Herculaneum and Pompeii nearby to the south of Naples. Pompeii is 40 minutes via the Circumvesuviana train Sorrento line, here: is the timetable (http://www.vesuviana.it/Reteeorari/Ferrovia/Orario/LineaNapoliSorrento, click on the station where you are getting the train) from the Naples Central train station.
From Pompeii, take a bus to Mount Vesuvius, and hike to the summit. Mount Vesuvius is the only active volcano on the mainland of Europe.
Paestum, an hour and a half to the south, near Salerno is Italy's most famous Greek excavation site.