Some of my favorites:
El Yunque National Rain Forest, Puerto Rico - tons of hiking trails, waterfalls (yes, you can get in them), etc. So exquisitely beautiful, cool even in the middle of summer, and it's the only rain forest in US territory.
Ocracoke Island, Cape Hatteras National Seashore - you can drive way out on the beach, far away (seemingly) from everything, camp, build a fire, enjoy nice wave action, and not have to see a bunch of beach houses or highrise condos. My favorite, true, beach experience.
Old Rag Mountain in the Shenandoah National Park - another beautiful hike with amazing views. Used to drive out for day hikes during the Fall when I lived in DC. Nice way to get up close and personal with the colors of Fall.
Brasstown Bald Mountain, GA - Again, lots of hiking trails around and up the mountain. Love exploring Brasstown Valley (Hiawassee, Blairsville, Young Harris) and all the surrounding mountains and streams. Good camping, water sports on Lake Chatuge, etc.