I'm the president of the local running club. There are a lot of good places to run. If you like trails, Croft State Park is great (single-track and horse trails), though it costs $2 to get in. You can get a lot of miles in there.
Milliken (hosts a lot of XC meets) is a nice place to run on grass. Spartanburg High School has a mondo track. Neighborhoods to the east of downtown (near the Rail Trail) are nice to run through (Converse Heights, Duncan Park).
Shoot me an e-mail if you want to know anything else. I could join you on a run and show you around, if you're interested.