Have 3 nightd to spend getting from DC to Nashville by rental car (public transport?) Any good recomenmdations for the route? Kentuky ? Any help welcome , thanks

I-66 west out of DC to the Front Royal exit. Travel south on the Skyline Drive as far as Waynesboro, VA. If you liked the Skyline Drive, continue on the Blue Ridge Parkway all the way to Roanoke. Get onto I-81 either at Staunton or Roanoke.
I-81 traverses the beautiful Shenandoah Valley if you choose the faster route. This is actually a two day trip with the overnight stop somewhere between Roanoke and Bristol Tennessee.

I agree.

You may also want to stop in Charlottesville and visit Monticello. Peaks of Otter Lodge would be a great place to spend the night.

excellent thanks. sounds good

Kentucky--Mammoth Cave(do wild cave tour), Corvette Museum, Corvette Factory

Charlottesville and Monticello would be good as would Shenandoah NP.