It's a bit away from Greenville, but... Calhoun Falls has some really nice campgrounds. The water is beautiful, and there are some really nice cabins. There are a ton of play areas for the kids, a really nice beach area and roped off swimming, several cabin rentals, several covered picnic areas with BBQ and pits, RV parking, a nearby store if you forget anything (really nice lady at the desk)... Just a beautiful area. There is also a ton of other stuff to do outside of camping... Golf course, Greenwood is a short drive, historic Abbeville, Savannah River. I could go on.
The individual camp plats at Calhoun Falls are some of the best I've found in several states. Not the best, but definitely among the best.
For what it's worth, we keep a Palmetto Pass, which allows us to visit all of the state parks free of charge, and have visited nearly all of the inland state parks. We are missing a few, but have been to ALL of the Upstate parks. Calhoun Falls, so far, takes the cake.
FYI, there is a Blue Hole recreation area just before/after the entrance to the state park that you may wish to avoid. It's a public loading and play area that is not controlled by the rangers.