In Nashville we like outside of town - out by the Grand Old Opry. My favorite place to listen to some music is The Palace and right next door is the (free) Duke's of Hazard Museum. If you are looking for something kind of cool & different to kill a couple hours go into the Gaylord Opryland Resort & Convention Center - it is very neat with a little riverboat & river throughout the inside courtyard area. The "real" riverboat is also a nice ride and that docks right in the mall parking lot at the far end from the Opry & Opryland. There is also tons of stuff to do in town but that is a lot of walking with little ones.
As far as Chattanooga we have not been their enough to give you great suggestions so I will let that up to others.
In Gatlinburg - Ripley's has the aquarium but you can buy a pass for several of the Ripley's attractions at a discounted rate. I forget what all of them were but they have 6 or 7 things total in the area. You can spend over a day just doing the Ripley's stuff alone. I actually like Gatlinburg much more than Pigeon Forge - to me Pigeon Forge is basically just an extension of Dollywood.