Regarding places to go backpacking near Chicago: yep, there basically aren't any, but I can offer one exception from experience - the South Kettle Moraine State Forest section of the Ice Age Trail in WI.
It'd probably be about a 2 hour drive from Des Plaines (it took me about hour and a half from Lake Zurich), and it makes for a fun weekend trip. Just a few complications - it's a linear trail, so ideally you'd have someone you could shuttle with, and advance reservations are required for the backcountry sites (which are shelters, so you don't really even need a tent). Also, you technically aren't allowed to backcountry camp anywhere else, so it's near the shelters or nothing - if I remember correctly, you are limited to the 1/4 mile area surrounding the shelters.
I was out there in mid-April. Car-camped Friday night, parked one car at Highway 12 Saturday morning, then drove to trailhead and hiked from near Shelter 1 to Shelter 3 (about 19 miles), stayed overnight (and experienced one hell of a storm) and then hiked out to our car at Highway 12 (less than 5 miles from Shelter 3). As for the scenery, my expectations were low, but it was actually pretty nice.
Good luck with the gear list, I'd try to help, but I think everyone else has said what can be said already.