Taking a public bus to Paestum
The train really makes so much more sense, but in the interests of being complete:
Hourly FS Bus Italia Campania (ex. CSTP) buses (#34) leave Salerno from Piazza della Concordia at the waterfront—a few blocks down from the train station (and a few blocks over from where the Amalfi Coast SITA buses stop)—and let you off at the SE corner of the archaeological site (60–80 min.)—one stop after the Paestum train station. Departures on this Line 34 run are hourly from 6:30am–12:30pm, then at 1:50pm, 2:50pm, 3:30pm, 4:35pm, 5:45pm, and 7:30pm.
There are also four daily buses from Naples (7:30am, 8:10am, 1:30pm, and 9:10pm); that ride takes 84 min.