I have an 8-year-old and a 5-year-old, and their favorite place in the world is City Museum. City Museum and the Arch are the two St. Louis things that can't be replicated anywhere else.
However, while there are zoos and children's museums and botanical gardens in other cities, the ones in St. Louis are really great. If you are on a budget, admission to Grant's farm and the zoo are free, you'll just have to pay for parking.
The Museum of Transportation is another favorite in our family:
Transport Museum Association
and if you have a kid that loves wooden trains, a visit to the Whittle Shortline Railroad in Valley Park is fun, especially on a too-hot or rainy day:
Whittle Wooden Train
They now make the trains in Louisiana, MO, but the store in Valley Park has a giant train table and several smaller ones where kids can play for free, and lots of toy trains and train merchandise for sale.