Mammoth Cave National Park
Been there, done that, got the t-shirt. If you are in the area it is worth a visit. I spent an afternoon at MCNP while in Nashville on business, driving over on a whim on a weekend just to get out of Nashville for a while. I took the Niagara tour which was enough for me; we have several cave systems here in Northern California so I didn't feel too compelled to spend additional money on more cave tours.
There are many different tours to choose from at various prices and ability levels. Summer is the busiest time of year; winter is slow and there can be snow on the ground. The caves maintain a year around temperature of 54F (12.2C) so most visitors wear a sweater or sweatshirt.
It is an easy albeit somewhat boring 90 minute drive from Nashville via I-65.