It's a nice park. Not hard to find. Entrance is right on 17, close to Brookgreen Gardens just across the highway.
When you get there, go to the visitor center and check in. We were there for a weekend in late 2011, and the rangers were very friendly and welcoming. Ask about activities for kids. You can get wifi there, sitting outside even when they are closed.
Distance to the beach depends on your site, seems like there were a few board walkways from the campground leading via a short walk through the trees to the sand dunes. Beautiful, very old live oak trees and an unspoiled beach. Take bug spray.
I don't recall cooking on a grate at our campsite, though they may have them. Seems like it was a ground level fire pit. Rangers sold bundles of wood at the visitor center. We put down one of those plastic RV rugs and that kept sand contained. But had to give it a good shake before leaving.
People were friendly when we were there, walking dogs and chatting.
The old mansion Atalaya is interesting to walk through. It is almost in ruins and unfurnished, right beside the ocean. Once quite a place, apparently.