Talladega GP is probably the closest for you to go ride on for a reasonable rate. It is a couple of hours away from Nashville I believe. It is about 3.5 hours from here. There are alot of track days and club races run there.
Supposedly the new Nashville Superspeedway has a decent infield road coarse built with motorcycles in mind. I have yet to hear of any bike running on it though.
A nice facility is being constructed in Birmingham, AL. I will be glad when that one is done. It is going to be a world class facility according to the rumors.
Road Atlanta is about 5+ hours from Nashville and since Panoz bought it the rates for bikes has gone way up but it does host the AMA Superbike series so it is the closest to Nashville for watching the pros. Mid Ohio shouldn't be too much further but I don't have a mad near buy. It is near Lexington, Ohio(I think that is the town). It is one of the nicest facilities that the AMA runs at.
The next track-day is scheduled for August 18, but I understand they are having trouble getting enough riders to commit to that date so it may cancel.
The Barber Track (near Birmingham) is going to be a great track with plenty of events (hopefully, and it may take a while for events to be scheduled there) and I understand that the local sportbike riders club, the Floribama Riders Floribama Riders has some big muckity-muck of the track as a member, so many, MANY track-days are expected to be held there.
It might not hurt to join the group (blatant commercial I know, but I am a member and they are a great group of people) and they are looking for someone to get a TN chapter going, anyway.