If you are at the Arch -
Jefferson Barracks (nice variety/nice length/medium difficulty) is about 15-20 minutes South. There usually are people at the course all day every day, so you should probably be able to find some locals to hook up with. Widely considered the most popular course in the area. Should be scorecards with maps in the kiosk. If not, here is the link to the scorecard and map...http://www.dgcoursereview.com/course_files/48/78d5d770.jpg
Sioux Passage (big length/mostly openish holes/some nasty tall grass/hard difficulty) is about 30-35 minutes North. Its a bit out of the way, so you might be all on your own unless you set something up beforehand. There should be scorecards with maps in the kiosk. Of the established courses, this is usually considered the Crown Jewel of STL. If not, here is the link to scorecard and map...http://www.dgcoursereview.com/course_files/63/02b0176f.jpg
Logan University (huge variety/nice mix of length/several epic hole designs/tough difficulty) is the newest addition to the area. It is about 25-30 minutes West. There has been a fair amount of traffic since it opened up, but you never know. The natural teepads can get a bit sloppy, but I think they should be frozen solid tomorrow. The course is a little tough to navigate with some longer walks here and there, but the design more than makes up for it...AWESOME! Here is a link to directions, scorecard and map...http://www.dgcoursereview.com/course.php?id=6521
Hope this helps!