Schwetzingen is in Baden-Württemberg.
As a tourist, walking is your best bet. The town is small and the 10-minute walk from the train station to the gardens is along one of the main roads, and on some days includes a market.
Thompkins Barracks is sort of within walking distance of downtown Schwetzingen. The walk is easy, but can take quite a bit of time. For those who would rather not walk, you can take bus 710 from one of the bridges adjacent to Thompkins Barracks. The ride will cost roughly 50 cents.
Visit the pubs around Schlossplatz (the place in front of the castle). If you like dancing, see the Cubar where you can dance salsa.
In May and June, you will get Spargel (Asparagus). Schwetzingen is famous for its awesome Spargel. Try it in restaurant Kaffeehaus, or Cafe Journal at the Schlossplatz.
If you like beer, try a Welde Pils with its award winning bottle design.