I made this trek at least 100 times in the long spell I lived up that way.
There are two main routes . . . .up through St. Louis or the further east trek through Memphis.
The St. Louis route has some highlights, specifically:
1) You pass through Tulsa in the early phase of the drive, so you can check out what that town has to offer (which is not much.)
2) Meremac Caverns south of St. Louis off I-44 . . a poor man's Carlsbad
3) the Gateway Arch National Park on the Mississippi. You can make a night of it in St. Louis, catching a cards game or eating at some of the better restaurants in town.
4) It takes you through Springfield, IL where you can visit the Abraham Lincoln stuff. They have the presidential library and museum as well as his home. Both are fun.
All of the above being said, I found the drive to be crushingly boring for most of it. The stretches between Tulsa and St. Louis are just boring rural nothing to do towns. The Southern Illinois stretch between St. Louis and Springfield is also soul-sucking.
The I-30 to Memphis route had prettier scenery and became our go to route.
1) You pass by Hot Springs and all that has to offer
2) Little Rock has the Clinton Presidential Library
3) Memphis is an awesome town - Beal Street, Rendezvous BBQ, Graceland to get your Elvis fix, watching the ducks at the Peabody hotel, the largest Bass Pro Shop in the world, awesome jazz and blues
4) Garden of the Gods Park in Southern Illinois
5) Site of the Lincoln Douglas debates around Cape Girardeau
6) Champaign, IL - Home to the University of Illinois. Eat at Joe's Brewery.