The St. Louis zoo is excellent. The arch is cool but be prepared to wait in line, although maybe you can reserve a spot on the web; I don't know but I'm sure a FRC team can do that research.
If you go to the City Museum and don't have a good time you should probably have someone check you to see if you have a pulse. Even if you don't like to climb around, just looking at all of the different things they have in there is quite entertaining. The biggest problem is having enough time to do everything.