We like
James Island County Park campground. It may be difficult to impossible to get a reservation there if you're trying to go in the next three months, though.
Our favorite thing to do in Charleston is to just go to the historic district and wonder around, eat at different restaurants, etc. One of the carriage tours is nice to help familiarize yourself with the area. With teenagers along you might consider Patriot's Point, a visit to one of the plantations or a drive out to one of the area beaches (Isle of Palms or Foley Beach).
