This spot has a ton of sites and since it is situated near an Amphitheatre if you plan ahead you can camp here and walk to the show! If you are looking for quiet, check the show schedule at The St. Augustine Amphitheatre though. 

The park offers the beach and a nice area of water known as Salt Run (good for paddle boarding or kayaking) there are hiking trails in the dunes and Downtown St. Augustine is about ten minutes away.