I have nothing to add to this thread.
But do be sure to look at www.roadsideamerica.com
Yeah, I was using the famous Midwest self-deprecation. I realize St. Louis can’t quite compare to a lot of the big cities in the US, and it has a number of serious problems (decaying urban core, racial divisions, etc.), but I really do love it. We do have a lot of great things here. In addition to what you mentioned, we have some cool neighborhoods, a terrific free art museum and free science center, Forest Park, some really great restaurants, America’s “other Little Italy”, and lots of other stuff. And we have a pretty good baseball team. Most years, anyway.
ETA: I went to college near Kansas City and lives there a while afterwards. I like KC a lot too.
Jewel Cave will be the most memorable part of a trip to the area. It is the best cave tour I have been on (out of ~15 around the world).
Personally, I would take the interstate up to Vermillion, SD, (excellent musical instrument museum at the university), and then head west to Pickstown and then take 18 or 44 through the Reservations to the Badlands. The reservations stick out like a sore thumb on any county-by-county representation of politics/prosperity/voting/education/population density for the U.S. and you’ll see why. And I would encourage your spending more time into Wyoming than “wasting” it exploring Missouri/Omaha. It really isn’t much different than STL. The difference comes at Pickstown, Winner, Chamberlain when you enter the west.