A nice trip in southern Missouri combines canoeing/kayaking, backpacking and caving. You can float the Eleven Point River (National Scenic River) that borders the Irish Wilderness. The White's Creek trail traverses the wilderness (small relative to most western wild areas, but large by midwestern standards) and it passes by White's Creek Cave. The cave is closed at certain times due to a Gray bat colony, so you need to check the USFS website for specifics. I have made this combo trip (4 to 5 leisurely days) solo several times, and the solitude and adventure are hard to beat!
Southern Missouri and northern Arkansas (the Ozarks) have some decent wilderness trails, karst landscapes, numerous caves, many very large springs (check out Greer Spring on the Eleven Point!)and gorgeous spring-fed crystal clear rivers. The St. Francois Mountains in SE Missouri have some fairly large tracts of public land (National Forest, state parks, state conservation areas, etc.)in an area with granite "knobs" and glades, deep forested valleys, great trails and few people.
Go to the National Speleological Society (NSS) website to find Grottoes near you.