You are sort stuck with I-81 for most of that trip, at least in the less populated areas. Once you get to Virginia, you'll go through or near a number of state and federal parks. The Blue Ridge Parkway and the Skyline Drive run roughly parallel to 81, to its west. They could not be more scenic, but the pace is slow. Along the route, some towns to see are Bristol, right on the VA/TN border. In southwest Virginia, there is something called The Crooked Road. It's a marked route through SW Virginia with stops along the way related to the music that has come out of the region. Smaller roads, quite scenic. Then when you are going thru central Virginia, I suggest you leave 81 around Roanoke and take 460 East to Lynchburg, where you can pick up Route 29 north. (Appomattox would be a slight detour but worth it.) Then take 29 up towards Charlottesville. Along the way you can stop at Poplar Forest, Jefferson's less famous house and Natural Bridge. Once you get up near Charlottesville there will be many scenic vineyards and lots of horse country. The vineyards are quite good and a nice place to spend an afternoon. Charlottesville itself is a great place - the downtown is full of great restaurants, etc, and UVA is really beautiful. From there leave town by Route 15 or Route 20 and you'll see nice scenery and historic things along the way. Fredericksburg is nice to visit. Or if you go north first through Orange County and any Route other than 29, you'll have a nice drive. 29 north of Cville is strip mall after strip mall. West of Charlottesville there are tons of horse farms. Check out Carters Mountain Orchard just outside Cville, en route to Monticello. Great view and you can get good and local cider and wine. Have a nice trip!
If you utilize Google Maps, it will suggest I-40 to I-81 north to I-66 and into DC. That might be the fastest driving route, but it won't give you any chances for trails on any of the interstates. I don't know much about TN horse trails, but I've lived and worked in VA for a number of years and one good resource you might want to look at for horse trails would be; https://www.dcr.virginia.gov/state-parks/horse-camping-trails The State Parks in VA are open again, and that link reference will give you quite a bit of information. You could consider driving over the Blue Ridge at New Market and looking for horse trails in that area and then you could head north again on US Route 340 which would bring you back to I-66 at Front Royal . That would give you a nice drive through some scenic and less crowded areas of Virginia. Hope I've been able to give you some useful info.