nelly,
I had a brief look at the TrenItalia web site and they indicate (for example) a train departing Roma Termini at 06:45, arriving Salerno at 09:16 (EuroStar Italia, ESI 9371, reservations compulsory). The next connection departs Salerno at 09:44, arriving Paestum at 10:15 (R 2427, second class only). With a travel time of 3H30M each way, it's probably a good idea to leave Rome fairly early. I didn't check on the times for the return visit.
Another possibility (depending on your budget) would be to travel to Sorrento (train to Naples and then Circumvesuviana to Sorrento) and then use a private Driver to get to Paestum. As I recall, there are a few listed in Rick's Italy book.
Happy travels!