I live here.
If you don't have to go through the city, don't. 294 is usually fairly quick. It's toll, but not unreasonable as Illinois tollways go.
When I pick up at a Northern suburb and have a run to Battle Creek, Michigan, Garmin will invariably try to route through the Loop and down to the Skyway. A slower, more expensive route could not exist, so I avoid that at all costs. Taking 90/94 through the Loop gets you into some of the heaviest traffic Chicago has to offer, and the Skyway-- Interstate 90 going into Indiana-- is so unbelievably expensive that it's in a class by itself. It's $5.00 just to get from the Dan Ryan to Portage, Indiana. I can drive from Beloit, Wisconsin all the way down to the Loop on 94 for less money than that. Now, these are 4-wheeler rates, just so's you know. Using I-Pass.
If I'm a bit farther West, needing to go from North to South or the other way, I-355 can be a good bet. It's a bit more expensive, but it is reasonably fast.
There is no non-tollway that doesn't take you through the heart of Chicago, and most can be a horror show during rush hours. Interstate 55 is probably the worst, it takes forever to go a relatively short distance. After that, 290 and 90/94 jocky for position as roads to steer clear of. There's no good non-tollway, I'm afraid.
Note: When I'm coming back from Michigan, Garmin always tries to choose the Skyway, the Dan Ryan and 290 as the route. I always over-ride it and use 294. The Skyway is expensive, and nobody sane uses the Circle interchange if he can help it. You'll be there FOREVER!!!!