In St. Louis, I've stayed at the Drury Inn at the Arch a few times. Nice hotel with secure parking. You can walk across to the arch. If there is a game, you can walk to the stadium. Do not stray too far, things can get rough quickly.

In Arkansas, nearly everywhere you go in the NW third of the state is beautiful. If you're an outdoor's person, you'll be in heaven.
The state puts out a motorcycle riding guide listing popular routes, destinations, and events. Their outdoor's guides are very good too.

I've been hearing good things about the Crystal Bridges Museum in Bentonville. Planning a weekend trip this summer to check that out myself.

One head's-up. In southern Missouri and northern Arkansas there is no such thing as a straight road or direct route. Travel times can be longer than you would expect.
They also don't have much money to spend on signs. If you see a double warning such as "hill" and "curve," take it seriously.