My sister-in-law lives in St. Louis, and we've been there often. There are a lot of great things to do with the family there: The Zoo, the Science Center and Grant's Farm are all excellent and free! The Magic House is a great children's museum that you won't want to miss - it's not magic shows as the name may imply! The Transportation Museum is great, mostly trains that you can go inside, but there are some old cars & trucks there as well.
The Gateway Arch is probably the most familiar thing about St. Louis, and is worth checking out if you're not too claustrophobic. There is a museum at the bottom of it with some Lewis & Clark and other historical displays.
The Hill is an Italian neighborhood, and there are some great restaurants and stores. Rigazzi's is where we ate, and it was excellent.
Chuy Arzola's Mexican restaurant has some of the best fajitas that I've ever had, definitely worth seeking out.
You've got to go to Ted Drewes frozen custard, it's a St. Louis indigenous staple.
Crown Candy Kitchen is worth searching out. They have an old fashioned soda fountain with some great ice cream. It's a good place for lunch as well.
We like eating hamburgers at Blueberry Hill on Delmar. Chuck Berry still plays there regularly. It's right across the street from Vintage Vinyl, one of my favorite record stores in the US. It's a really cool place.
Union Station is interesting. It's an awesome old train station that has a mall and hotel in it now. Go upstairs and check out the hotel lobby, it's really beautiful. Check out the archways just inside main entrance and try the acoustic tricks that con men used to use (there's a sign explaining how to do this). It works, and it's really cool.
There are tons of things to do in St. Louis, but those are the most family-friendly things that we've found and enjoyed. Of course there's the Cardinals baseball, I guess that's kind of obvious, though.
Some links:
http://www.stlzoo.org/
http://www.slsc.org/
http://www.grantsfarm.com/
http://www.magichouse.org/
http://transportmuseumassociation.org/
http://www.gatewayarch.com/Arch/
http://stlouis.missouri.org/thehill/
http://www.saucemagazine.com/rigazzis/
http://dine.com/restaurants/rid/61437/index.html
http://www.teddrewes.com/Drewes.asp
http://www.crowncandykitchen.net/
http://www.blueberryhill.com/
http://www.vintagevinyl.com/
http://www.stlouisunionstation.com/
