Ah, so you should be somewhere around the 22nd and Cermak area or not too far from there.
There are three major tracks around here that people go to.
Route 66 down in Joliet which is 50 minutes south or so from where you are. Nice drive there, toll free, just 94 east to 76 towards Iowa, exit 132 or something and then down the road for 3-4 miles and it's the second entrance on the left past the oval track known as Chicagoland Speedway. The track itself is very nice, highly professional, but from what I have read and heard from many racers, the tech inspections are really tough, runs are limited, and they hold very few test and tune nights, and when they do, they're not exactly all that cheap. They can be reached online at
http://www.route66raceway.com
Great Lakes Dragaway in Union Grove, WI, is about an hour and a half straight west/north on 94 and it's exit 337 I believe and then turn left at the stop sign and plow on down the road until you see it on your left maybe half a dozen miles down the road, tolls between your house and there is $1.25 total. The facilities are not nearly as nice as Route 66, but they literally have allowance for test & tune pretty much any week night and from what I understand, admittance is cheaper and tech inspection is not nearly as strict. The Summit Points series is quite good and very competitive. By far my favorite track to go to as it's the closest for me because I am about, ohh, 25 traffic-free minutes north of the 22nd/Cermak area. You can get more information at
http://www.greatlakesdragaway.com and their 2004 schedule is up already.
Last is Byron, and this seems to be about as popular as Great Lakes for test & tune because their tech inspection is basically non-existant on test & tune days and I understand it's the cheapest to run there. However, with 6 tolls total and by far the longest (in mileage and time) drive there, I like it least. Also, few people go that far to race, so if you mess up, likely few people you know will be there to chuckle.

Also, several times I have had problems with this, the printout timeslip does not match the clock time/MPH at all times, so there may be some confusion there. It's not off much, but off is off. Track information may be found at
http://www.byrondragway.com