Lone Elk Park is nice if you want to go for a little drive and want to try to catch a glimpse of elk and bison. Note: sometimes the animals are right up close, and other times they are no where to be seen. If you do drive out there, the World Bird Sanctuary is close by, interesting & free. I wouldn't necessarily make either place a specific destination, though.
City Museum is great as long as you are comfortable allowing the kids to be out of eyesight every so often, as there are passages and crevices that are too small for most adults to enter (but that kids obviously love).
The Magic House is always fun--in fact, when my nieces & nephew came out to visit, they felt it was the highlight of their trip (ages 7-11). Also, check the Magic House website for possible special events during your visit. If you are members of a children's museum at a high enough membership level, then remember to bring your museum card for free or reduced admission. I think it's called ACM reciprocity? If you don't have a reciprocal membership, be on the look out for buy one/get one coupons, or for a groupon.
The Science Center is another place that has lots of great stuff. The Science Center entrance is free, but I think you can get reduced prices on the Planetarium and the Omnimax if you have a science museum membership elsewhere. You don't have to park in their paid parking lot if you don't want to--there's free street parking near the main entrance, and free parking on the Forest Park side (near the Planetarium).
Depending on the weather, the Botanical Garden can be really great. Again, they offer reciprocal privileges with many other botanical garden memberships. My kids love the Children's Garden there, but it may not be open when you are here (I think it usually opens in late March or early April? costs a little extra on top of admission).
If the weather isn't too bad, the zoo is another great place to go! There are many indoor exhibits, so we have popped in and out of buildings and been fine. Free admission but also reciprocal privileges with many other zoos for reduced prices on things that have a charge. If you go the first hour the zoo is open, then some of the things that have a charge are free (such as the children's zoo or carousel). There's free parking on the streets in Forest Park, but the specific zoo parking lots are kind of expensive.
The History Museum (also free admission) has a new-ish interactive exhibit area that is directed at kids called History Clubhouse. We haven't made it out there, but I know several people who have enjoyed it.
Oh, and if anyone likes history, the Ulysses S. Grant National Historic Site is very interesting--there's a historic house to tour, and information/exhibits on the Dent family and the president. I would say it's more of an adult attraction than a kid one, but my kids still had fun.
If you are interested in driving a little ways, you can go over to Alton, Illinois. The bald eagles will probably have already left the area by the end of March, but if they haven't, watching them fish in the Mississippi River is an awesome sight. You could also visit the National Great Rivers Museum/Melvin Price Lock & Dam, where you take a tour of the lock & dam, and can watch boats and barges go through. I want to say there is a minimum age for the lock & dam tour, but I can't remember what that age is so maybe there is not. Also free!
Another slight drive away is Shaw Nature Preserve. It's affiliated with the Botanical Garden, and there's some great hiking out there. There are thousands of daffodil bulbs that have been naturalized there, so it's really gorgeous in the spring.
Finally, we haven't stayed at Timber Creek yet, but we're actually going to go there in a few weeks for a local "country" getaway.
Hope that helps! Let me know if you have any questions!