My personal opinion (I live right outside of Bristol. I go to Bristol/Kingsport/Johnson City often)
The way I go is 81 to the first Tennessee exit (79 maybe? It has the Basspro/Pinnacle) and take 11w to kingsport. More lights and stuff but less chance of an accident. 26 is terrible and notorious for wrecks and reject drivers.
I get off 11w and hit 23 and take it up to Diffield, then hit 421 and hit the steep hill in Lee county/Pennington Gap.
You CAN go straight through Duffield (don't turn at McDonalds/gas station) and head up to Big Stone Gap and take 58 over and bypass the big grade coming up by the lookout, but it's a bit out of the way (around your elbow to reach your ass).
If your truck doesn't like a load or you don't want to go 30 up a good climb, I'd go that route.
However, a 2014 Tundra 5.7 with a good load pulls it fine.
Either way, you'll come to 58/421 in Pennington Gap at the red light by Food City. Get beer/groceries here or the gas station up the road where you turn right. Best prices and Harlan is a dry county. That's pretty much the last stop before you go up then down the mountain into Harlan.
TLDR: 26 sucks (think Asheville inter-fucktion and wrecks daily). I take 11w from 81 to Duffield, hang a left and hit 421. Go to Pennington Gap and get beer/food/charcoal/beer etc at Food City and roll towards Harlan.
Up to Big Stone Gap is less hilly but longer (30 mins to hour).
Also, if anyone is up to stop at the Pinnacle, I can hook you up food wise.