I know this thread is a couple weeks old, but I'm in the northwest suburbs of Chicago as well.
Badlands is great; a few hours there and a few hours back means it's a weekend thing. There's a few hotels in the area and they've got on-site camping, so that helps.
Redbird State Recreation Area is also in Indiana, but a little further than Badlands. I don't remember if it's bigger or smaller than Badlands, but still a fun place to visit.
Cliffs Insane Terrain used to allow 4x4s. Unfortunately, to save the park, the new owners were awarded a grant from the state of Illinois that doesn't allow anything larger than dirt bikes and side-by-sides. It's a terribly outdated stipulation in the grant and the owners were (unfairly) griefed on the park's Facebook site for using it. Had they not received it, though, the park would've been shut down.
Every state has its upsides and downsides. Do your due diligence.