Having made this trip several times:
Unless you are adamant about Skyline Drive, I'd travel thru the Shenandoah valley (parallel to I-81) and get on the Blue Ridge at Waynesboro. BRP is 469 miles of wider road with so many scenic things to see that the time to travel depends on how many things you want to see.
I did it last Oct coming home from Wingstock in two days. Photos are on my website. As far as places to stay, Peaks of Otter & Little Switzerland are on the BRP and command a bigger cost. Being the frugal traveler, my half way stop was at a Comfort Inn just off the BRP in Roanoke, VA.
However, knowing where the best food is, there is a great family style restaurant just off of the Fancy Gap exit, to the west side of the BRP. Don't miss it! Gas is also available.
Before I went, I did a Google search for all websites, printed out the gas stops close by and made sure I downloaded all the maps.
This is a trip that can consume two days or two weeks - just depends on the number of stops. Enjoy the ride.