We just went to Charleston a couple weeks ago. James Island County Park, right in Charleston, was phenomenal. We had a blast. During the summer months, there's a little water park right on site with a lazy river and such. We rented bikes and rode and rode and rode. There's an insanely incredible dog park right there, too, if you've got a dog.
Here's a video from our trip. The park is perfectly located so we were within minutes of downtown, the aquarium, Patriot's Point, the works. Loved it. If you go -- let me know what you think!
We stayed there last year and it was great! Going back again in August! It's also very close to Folly Beach.
We also stayed at Oak Plantation Campground in Charleston a few days before going to James Island. Nice campground but getting in and out was awful as traffic was constantly backed up and a 4 lane road runs in front of it.
It's already been said, but, we love James Island County Park! Waterpark just outside the gate, around 10 min. from the beach, shuttle service to Charleston, three nice bike trails, huge playground/fountain area for kids, paddle boats, fishing etc. Did I mention very friendly atmosphere!
We tried to stay at James Island Park but they were booked up. But we did go by to check it out and it was really very nice. Excellent dog park there as well. Definitely next time!