Answering as a native Virginian ....
Shenandoah NP: I've stayed in all NPS campgrounds in the Park over the years and you can't go wrong with any of them if you can do without hookups. BUT, the park is EXTREMELY busy that time of year. Skyline Drive traffic is heavy and campgrounds are packed. So make your CG reservations well in advance and allow plenty of time on the Drive. (Allow time anyway throughout the year, for that matter; the speed limit is only 35 mph.)
In fact, expect busy-ness in all of the mountains of Virginia during peak fall colors.
The only private campground near the SNP I've stayed in is the
Harrisonburg KOA. Good experience.
I assume you've checked out the Blue Ridge Parkway connecting SNP with the Great Smoky NP. I've never stayed in any of the federal campgrounds along the way but I've camped in a number of Virginia state park CGs. In general, Virginia does a pretty decent job with its campgrounds. More elbow room between sites than in most private campgrounds, well maintained facilities, clean restrooms (usually), courteous staff.
Another private campground I can suggest is in
Montebello, not far off the Parkway. A few miles down the mountainside from there is
Crabtree Falls, which offers various views of the falls via several hiking options, from a short paved handicap-accessible segment to a rigorous loop.
As part of your driving loop, consider central and eastern Virginia. Certainly not as scenic as the mountains, but lots of history if you like that. A big negative to that, however, is traffic in the I-95 corridor as you head to or from PA. I can try to answer any specific questions about this area if you're interested.
As for hiking: many possibilities for all skill & endurance levels. (I hiked Hawksbill, the tallest peak in SNP, just last Saturday. It's rated moderate in the NPS park literature, though for this old-ish and overweight-ish body, I'd rate it strenuous.) The Appalachian Trail more or less parallels the Skyline Drive and Blue Ridge Parkway in Virginia, with countless side trails to choose from instead of or in addition to the AT.
If you like music and its history, especially country, consider some or all of the
Crooked Road in SW VA.
Enjoy! And spend plenty of money in VA!