Welcome to RTA!
Answering the bit about Mammoth Cave: yes, it's worth it, very unique among caves since it's so.....well, mammoth sized (huge). There are several tours available. At most national monument and park caves, you go on a tour so they can explain what you are seeing (and keep you on the safe path).
It's your trip, so some of your questions are only going to be able to be answered by you, dependent on how much time you have. Are you on a timetable or strict budget? If you have the time, wandering through the Ozarks is a great idea. Avoiding Chicago, if you don't want to go there, should be easy enough.
Are you planning this from paper maps or a road atlas? If not, it's time to go get some maps or an atlas -- if you're AAA, of course you can get some from your local office.
Donna