A surprising number of European tourists travel Old Route 66 from Chicago to Springfield, Illinois as a day trip. Some of the bigger towns along the way make a special effort to welcome them. There are several organizations that provide information.
If you want to avoid the more heavily trafficked areas, pick up a map in the visitor's center, then drive to the east side of the park.
Some lovely canyons and waterfalls out there away from all the hubbub. And check out Matthiessen State Park as well, it's basically a mile long canyon you can walk in.
But what about a true day trip/hike is Devil's Lake State Park. There's some really cool trails along there, and some awesome rock formations that are usually occupied by true climbers.
Definitely worth a trip, and you can hike from 5 miles to 20 miles before heading back to the city in the same day. It's about a 3 hour drive, and you could stop in Madison for the night (it's the capital city of Wisconsin).