I used to live in the St. Louis area. Folks have already given you a list of some of our favorites (the zoo, the science center, Magic House, Grant's Farm), but I'd like to add one. The Missouri Botanical Garden (the do charge admission, but I remember it as being reasonable), which is in St.Louis, is an amazing place to go with children. In some parts you do have to keep to the path, but there are also areas where children can run. There is a children's garden, but my favorite is the Japanese garden.
For food, St. Louis has an Italian quarter where you can eat amazing food for reasonable prices. Someone has already mentioned Drewes frozen custard (YUM!), and I agree - you simply must make time to go by there. Another kid-friendly St. Louis specialty that you can find many places is "toasted" ravioli. It's not really toasted - it's fried raviolis that you dip in marinara sauce. Not exactly health food, but very, very tasty.
I do not know if it's still there, but there used to be a butterfly house that was a fantastic place to go with children, as long as your kids don't fear flying insects. You get to wander through the butterfly habitat, so there is a possibility of a butterfly landing on you.
Do be aware that there are some parts of St. Louis where you would not want to be with after dark. The folks at the time share could probably tell you specifically which areas of town to be careful in or to avoid. To the best of my knowledge, the areas around the major attractions are reasonably safe.
Outside the city in Illinois is Cahokia Mounds, which another poster mentioned. While my kids found it fascinating when they were older, when they were the ages of your kids, they simply weren't interested.