+1 for StonesRiver on the rifle side. I'm still not comfortable with the way the handgun side is setup, but I'll go occasionally, like this afternoon.
About 45 minutes south of downtown is CHMR. It's a $40 membership for a year. You can go every day that they're open all day long and you're still only out $40 for the whole year. It doesn't get cheaper than that and you'll never find a nicer guy than Charlie who owns and operates it. Middle TN TGO meets are often held there. It is good for rifle. He even has a couple of clay throwers setup for shotgun. There is also a small handgun pit. Rifle distances are set at 25y, 50y, and 100y. I hear he's working on some 200y lanes, but I don't know if those are finished yet.
CHMR
Guns and Leather in Greenbrier has a computer controlled indoor handgun range (5 lanes) with lots of programmable modes. I think it's $15 for an hour. They also have 5 lanes of just motorized target clips (the regular indoor range stuff) for $10 an hour. the Hendersonville store doesn't have the computer control if that's important to you.
For shotgun, nothing beats the setup at TN Clay Target Complex. But it's not cheap.
http://www.tnclay.com/