Paestum is a Greek settlement where you can find the best preserved Greek doric temples worldwide. Most of the ancient Greek city has not been excarvated yet. Besides three temples, the side has a gymnasium, city wall and tower ruins and plenty of house wall ruins. Part of the site is also a museum showing excarvated items such as statues, vases, metalwork, painted grave stones and can be summarised as well impressive.
There is really nothing else at Paestum except fields and fields and fields.
If Italy would have had a word in where to find such a site, it would have been somewhere else.
By Train: train goes from Salerno to Paestum station. Read about the unified public transport ticket Campania Unico. Buy your ticket to get out in one of the souvenir shops close to the museum or before you come here. No ticket sale anymore at the station and machine never works.