I grew up there and there's a lot more than has been mentioned. Great ideas have already been mentioned, but here are a few more:
The St. Louis Science Center is free and a multi-million dollar facility, but that might be better suited if you were traveling with children. It's in Forest Park near the Zoo.
Also in Forest Park and also free is the Art Museum and the Museum of Transportation. Both are pretty good museums, if you are the museum type.
If you like history, you might like to see the indian burial mounds at Cahokia, just over into Illinois.
There are some nice caves about 35-45minutes southwest of St. Louis. Meramec Caverns is the biggest and most commercial of them.
Six Flags-St. Louis is a good place if you like amusement parks.
Finally, (and MOST importantly)...Eat Italian. St. Louis has always had some of the best Italian food anywhere. Go to south city and eat on The Hill (my personal favorite is Cunetto's). Or in the north suburb of Florissant, I really enjoy Rizzo's (I-270 at Elizabeth). The other must-have for food is the frozen custard at Ted Drewes.
You could stay at some nice downtown hotel and get to several of these places on the MetroLink. It goes out to Forest Park, but beware of crossing the river from St. Louis. East St. Louis is risky at best.