Frankly, you are going to be disappointed.
Missouri national forest area (Mark Twain National Forest) is closed to motorized use unless marked open. This policy has been in place since the late 80s and the FS hasn't got around to marking anything open. There's trail to be ridden (if you can find it), but you'll need a plated bike to connect the trails together. There are public riding areas....
St. Joe state park (2000 acres) is close by (60 miles) and cheap ($3/day), but most of the trails are now atv wide.
The Midwest Trail Riders Association (MTRA, $10/yr membership) has a 600 acre riding area 110 miles from St. Louis. Nice single track, but 600 acres worth.
Finger Lakes state park is around 130 miles away and is truly a quad heaven.
Chadwick - a 230 mile trip to an 11,000 acre OHV area run by the Forest Service. It used to be a great place to ride, but the quads have overrun that place too.
I'll mention Sutton's Bluff only to be complete.
This place is supposed to open soon.
http://www.sandycreekmx.com/ It will be around 60 miles from St. Louis (near St. Joe park).
Illinois has opened a couple of riding areas within a couple of hours of St. Louis, but I've yet to visit them. Little Egypt should get you a hit in google.
I practice at St. Joe park and get my riding fix by running enduros, the odd hare scramble and taking multi-day riding trips to other parts of the country.
Welcome (back?) to Missouri.