Here are some free things in the Chicagoland area. I did some poking around to make sure they were still free, since I moved away back in 2000. You'll have to go through some rougher areas to get to some of these places, but that's true if you go too far in any direction from U of C.
Free stuff:
Lincoln Park Zoo
Chicago Botanic Gardens
Lake Shore Drive is really pretty if you've got a car
all the beaches around Lake Michigan
The Art Institute has a free night once a week. Possibly it's Thursday but it's best to double check.
All the museums have free days. This site:
http://www.chicagoparent.com/picks/museum-free-days has a run down. I especially like the Field Museum and the Museum of Science and Industry. The science museum has this really cool international Christmas tree exhibit each year.
Grant Park and Buckingham Fountain are kinda cool to see.
Watertower Place
The Magnificent Mile with all the really high-end fancy shops
The Chicago River, especially when it's been dyed green for St. Patrick's Day. (St. Patrick's Day is huge there, btw)
The Harold Washington Library
There are a lot of really pretty Catholic (and other) churches throughout the city. Most are open every day.
Here are some other sites that list free stuff but I don't know much about them:
http://www.choosechicago.com/articles/v ... TIONS/420/
https://www.thrillist.com/entertainment ... in-chicago
http://www.fodors.com/world/north-ameri ... o#!1-intro
http://www.chicagotraveler.com/free-things-do-chicago
Other random things:
Get a picture with the Michael Jordan statue outside The United Center, or that weird Bean thing downtown (that was after my time)
Eat as much authentic ethnic food as you can. The food in Chicago is so amazing.
Watch them film a TV show/movie somewhere in the city. Sometimes they need extras, too, so that can be a fun experience if that's your kind of thing.
Try to pick up a Chicago accent. Start putting the definite article in front of everything, especially place names.
Watch My Big Fat Greek Wedding
Go past Midway airport. It's an airport, right in the middle of a neighborhood. Weirdest thing.
Volunteer with any kind of population that would make you a more competitive teacher wherever you move next. There's just about every kind of student somewhere in Chicago.
If I can think of anything else I'll let you know. I haven't lived there in a long time, and I'd never move back, but I still have a lot of love for the place.