Here is my recommendation:
Day 1- Arrive in St. Louis, pick up rental car, explore downtown (I recommend the science center), go to Cardinals game at 7:15pm
Day 2- Visit Six Flags St. Louis, drive to Branson
Day 3- Visit Silver Dollar City
Day 4- Drive to Kansas City, go to Royals Game at 7:10
Day 5- Visit Worlds of Fun, drop off rental car, fly home