The False Cape State Park (Virginia Beach) is in southern Virginia Beach. The Access is through the Back Bay National Wildlife Refuge Silvestre. The park offers camping and an environmental education program. You can also take the Terra Gator, a vehicle specially designed that minimizes the harm done on the environment.
Stretching 105 miles long, Shenandoah National Park is a massive protected area not far from Washington DC, Richmond, Charlottesville, or other Virginia areas. The park's famous road, Skyline Drive, runs the entire length of the park and provides spectacular views. Fall is a popular time to visit, in order to see the colors of the fall foliage, but winter can actually provide an even better experience since there won't be any leaves blocking your panoramic views into the valley.
Along with scenic drives, the park has a number of great hikes. The most popular - and my favorite - is Old Rag, a moderately strenuous day hike that involves rock scrambling, narrow ledges, and great views from the top. However, there are plenty of other options in the park that are shorter, easier, and/or less crowded. Head out for a part of the Appalachian Trail if you can.
Wildlife sightings may include deer, black bear, raccoons, and numerous birds.
Overnight stays are available through backcountry camping, designated camping spots for cars/RVs, and hotel-style lodges.