I second the recommendation for Mammoth Cave National Park, unless caves aren't your thing.
If you want waterfalls, you'll have better options staying in Tennessee. Burgess Falls and Cummins Falls are both about 1.5 hours east of Nashville, and can easily be combined in a single day as they are less than half an hour apart (just did that last weekend, about a 7 hour trip total). Burgess Falls is a BIG waterfall, among the most impressive I've seen in the Southeast, though you really need to go to the base to properly appreciate it.
Edited: 11:24 pm, September 25, 2012