Hey there, I'm from St. Louis too. We're actually going to be driving down at the end of April next year and I can't wait. Anyway, we usually stop in Atltanta, GA. It's about 10 hours from St. Louis to there and then I think another 8 hours from Atlanta to WDW.
Here's another couple of options that we have done and are going to try.
When we went a few years ago ds was 4 1/2 and dd was 18mos. We left at like 11:00 or 12:00 at night and drove through the night so the kids slept most of the way. We made it to Chatanooga around 10:00 or 11:00 and stopped and did some sightseeing and had lunch. Then we drove on to Atlanta and stayed the night, we got in around 3:00 or so. Now the next day we actually stayed in Atlanta and visited friends and then drove on to WDW the following day, but you could just head out the next day. It allowed us to get in early enough to get dinner and get the kids to bed at a normal time and we went to bed early to catch up on some of the sleep we missed.
Now this is what we are going to try next year. This time though ds will be 6 1/2, dd will be 1month shy of 5 and new dd will be about 15mos. Plus we will have a friend going with us who will help drive also. Our plan is to leave Friday around 6:00 or so and drive straight through to WDW. There will be three adults to take turns driving and napping and playing with the kids. If I plan 18 to 20 hours to drive we should get into Kissimmee around 2 or so in the afternoon. I figure we can find a cheap hotel to spend the night and get some rest after the drive and then head over to WDW Sunday late morning or early afternoon.
Whatever you do, good luck and have a great trip.