The Lost Sea is in Monroe County Tennessee, on Highway 68 between the little towns of Sweetwater and Madisonville, which are half-way between Chattanooga and Knoxville (just off I-75). They have a pioneer village, too (at least they used to). Easy to spend two or three hours there without breaking the bank.
If you and your kids are really adventurous, consider a drive along the Tellico River and jump/dive off of the Baby Falls (a few miles above Tellico Plains in Monroe County and just about a mile above the larger Bald River Falls). It is scenic, even if you don't brave the falls.
Either place (or both) is a good stop between Chattanooga and Gatlinburg/Pigeon Forge (Dollywood)
