Take Briley Parkway (SR 155) from I-65 around to I-24. It is not terribly far out, but avoids downtown. This is not an interstate level road but it is decent, with 4 lanes or more and not too many traffic lights.
US231 is a good road but it goes through every dinky little town with 18 traffic lights, it will take forever.
Alternatively... pick up I-74 in Indianapolis, over to Cincinnati, take the big loop around Cincinnati and pickup I-75. Take I-75 down to Lexington, Knoxville, Chattanooga, Atlanta, and all the way south. This may be a longer route, but it avoids major snarls except Atlanta.
Check
TDOT SMARTWAY TRAFFIC for current construction & delays. There shouldn't be any big construction that time of year. Mid-day is the best time to get through Nashville, or very early morning, as in before 6 AM. Avoid 7-9 and 4-8 headed that direction.