We went from Northern Illinois west of Chicago to Bristol twice a year for 15 years. Tried all the routes. The best we found is this:
I-80 to I-65 south; I-465 around the north side of Indy to I-74 east (good Cracker Barrel for an overnight stop at exit 113 - we used to leave after work and hit it about 1 AM); take i-275 around the SW side of Cincinnati to I-75 South; I-75 south to Corbin, KY - exit 29 - and East on on US 25E (watch the Corbin exit with an RV, it is a steep downhill on a short ramp; there are a couple of stiff climbs on 25E, but no big deal);
Through the Cumberland Gap tunnel on 25E (have your propane tanks turned off) then immediately turn onto US 58 east through Virginia. Good four lane with virtually no traffic all the way across to Jonesville, then a stretch of good, but twisty, two lane into Duffield, picking up US 421 along the way. Fairly long and steep downhill into Duffield.
Stay on US 421 out of Duffield (right turn there) and follow the four lane down into Gate City. US 421 turns left, signed to Bristol, but stay straight on US 23 towards Johnson City. After a short distance it turns onto a limited access four lane that becomes I-26 south. Take that to I-81 North, to exit 69 for Blountville. Follow TN 394 all the way around to the track area.
Easier driving it than describing it! Climbs are no problem with our current rig ( see sig ), but we also did it many times with a Ram 1500 towing a 7k fiver with no issues. On the return trip the climb West out of Duffield was tough - with the Ram hitting the bottom at 65 meant 20 at the top! The Volvo climbs it in 9th gear at 60 with a 18k fiver!
Most important, have fun! Bristol is a happening, including the four or five days before the race.