I have to agree with all the recommendations everyone has given. There is a Da Vinci Machines Exhibition that will be in St. Louis till May and it is close to the convention center. I haven't been there yet but have it on my list of things to.
http://discoverdavinci.com/
The History Museum at Forest Park is wonderful. Admission is free but there is a cost for special exhibitions.
http://mohistory.org/
Pappy's Smokehouse has great BBQ at a reasonable price.
http://pappyssmokehouse.com/
The St. Louis Zoo is always wonderful. Admission is free
http://www.stlzoo.org/
Hope you enjoy your time in St. Louis.