The best place to park is in the Grayson Highlands State Park, in the Backcountry parking area at Massey Gap. It's safer, and you'll spend a lot more time in the open high country. It costs a couple of bucks a day to park there. You pay at the park entrance station.
There are several loops that you can see on the map (get the National Geographic map of the Mt Rogers National Recreation Area.). The shortest combines the AT with the Pine Mountain Trail. That's about 12 miles including the blue blaze trail that goes up to Massey Gap and the AT and makes a nice overnight hike.
Two night hike idea: Park at Massey Gap, hike up the blue blaze trail to the AT, and turn left (southbound on AT, but compass NW-ish) to hike up Wilburn Ridge. At Rhododendron Gap turn left to stay on the AT southbound toward Thomas Knob Shelter. Head past the shelter a quarter mile and take the side trail to the top of Mt Rogers. No views, as it's in the woods, but it's the state high point. Come back down to the shelter, go past it (AT-north, back the way you came) anywhere from 500 yards to 1/2 mile and find a camp site. There is water in the spring behind the shelter (inside the fenced corral.) There is a nice privy at the shelter (chicks dig privies.) You can't camp at the shelter, but there are many, many obvious campsites between the shelter and Rhododendron Gap. Almost all of this is out in the open on a rocky bald ridge line. (Bring sun screen.)
On day 2, go back to Rhododendron Gap and take the Pine Mountain Trail over Pine Mountain. In 2 miles you'll reach the AT again. Turn right on the AT and hike past Scales and over the long open crest of Stone Mountain. When you re-enter the woods after Stone Mountain, you'll walk maybe 1/4 mile and find a small meadow to your left - it's an old homesite. There is a spring just off the trail at the edge of the meadow. It makes a pleasant camp site. It's not all that long a day for miles, but I always like to take plenty of breaks in the open country and just watch the sky and the wild horses. If you want to keep going a little longer, there are many more tent sites down near Quebec Branch (Creek) when you cross out of the national forest into the state park. Camp in the NF (no camping in the state park) -- look to the right of the AT just before crossing the last creek before the Wise Shelter. The shelter has a spring and a privy.
On day 3, keep going sobo on the AT past Wise Shelter and back up to Massey Gap and then down the side trail to your car.
There are ponies in lots of places, both on Wilburn Ridge and back near Scales and down at Wise Shelter. There is no real guarantee of seeing them, of course, but I can't remember ever being there without seeing ponies. My daughter first overnighted in this area when she was 4, and did her first multiday section hike here at age 7, so your kids should be fine.
After the hike, you can drive down US 58 west to Damascus, the best little hiking town on the AT, and hit the Blue Blaze Cafe for some post-hiking pizza. Also check out the outfitters, the coffee shop, etc. You can easily get from Damascus back up to I-81 to get home. There are motels in Marion, and of course at most interstate exits. There are also several B&B and hiker inns in Damascus if you want to spend the night there -- I'd recommend the Lazy Fox Inn. Makes a nice extra day, as you can get showered and get a nice dinner before heading home. If you can spend another night in town, then you can take the kids on the Virginia Creeper Trail, an old railroad trail that is now a very popular bicycle trail. There are a dozen different outfitters in Damascus who will rent you some bikes and drive you up to the top of the hill for the 17-mile gentle downhill ride back into town. Fun stuff.