I used to do a lot of camping around Greenville as a kid and I have a couple favorites.
Jones Gap is very fun, you can swim in the CLEAN river, feed the fish at the fishery, hike on the trails and enjoy their nice campsite.
Kings Mountain is about an hour out but it is worth it. They have great camp sites, great hiking and an awesome welcome/history center. Lake Jocassee is fun if you have access to a boat, if not you better know how to fish, cause that is about all there is there. Really Pickens County and Cherokee County, is just loaded with camping, hiking, boating and fishing. I hope I helped, good luck.