I've actually seen quite a few skate parks around. What's cool is that you can learn by observation and also I'm sure people there could show you a few tricks too. I'd be very surprised if skateboarding was allowed at Falls Park. Which, IMO, is kind of silly considering that bikes are allowed. I'd imagine that the clientele at the parks would be the best people to ask about where to skate in public.
As far as I know, these are the most popular ones in the area:
Skatepark of Greenville
(http://www.skateparkofgreenville.com/wp/ - broken link)
I have seen other pocket skate parks though in Greenville, I just don't know the names.