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.
There is a train service from Salermo that stops at the Paestum station. From here it is a 20 min. walk to the museum and at least as far to anything else.
Besides the historical site, the accompanying restaurants (cheap!) and souvenir shops (do all your Italy souvenir shopping here, including all the Rome and Northern Italy stuff because it is cheap and there is loads of stuff!) there is only a beach half an hour walk down the road. The beach itself is clogged with cheasy restaurants and souvenier shops mainly aimed at italians, even though one chap in the restaurant spoke German.