I live in Chicago, but occasionally make it back to The Lou and hike with my dad. Here are some cool spots within 1-2 hours drive of downtown:
Hawn State Park (I55 to 32): There's a 6-mile loop up Pickle Creek valley, then up and along the ridge through pine forest and over rocky outcrops to a side valley and back down. It is a great, great dayhike. There is also a backcountry campsite.
St. Francois State Park (I55 to 67): Stop at the first day area and follow the trail up the creek valley, then veer to right up the first side canyon, climb the straight line ridge to the south and pick up a trail heading west and back down into the creek.
Shaw's Garden Extension (I44 to Gray Summit): Really nice place about 40 minutes from Busch. Hike all the way to the Meramec.
Ozark Trail: Check out the Ozark Trail Association website for details about the sections (http://www.ozarktrail.com
). There's a great section called Karkaghne (I helped build it) that's near Johnson Shut-ins.