The Saint Louis Zoo is a great free way to spend the day. Entry is free, although some premium attractions like the Children's Zoo require a fee. However, the Children's Zoo is free from 9am to 10 am every day!
Lots of things to do and see, and there are lots of tables for that family picnic.
Parking is expensive, but if you don't mind walking, there is free parking nearby in other parts of Forest Park.