M.C.
The children's museum at Navy Pier is free for children under 1 yr - so you'd only have to pay admission for the adults.
Lincoln Park Zoo is great if the weather is nice. Wednesday mornings they have a sing-a-long in the yellow barn at the farm in the zoo. There are 2 shows - 9am and 10:15 (I think). It is loads of fun. $1 donations are suggested.
Lincoln Park Conservatory or Garfield Park Conservatory are free and beautiful. LP has a great holiday train display. I used to take my daughter to both conservatories all the time when she was an infant. She loved the colors and smells and its always warm and tropical in there.
Check the Chicago Cultural Center - they usually have something interesting going on, and call Borders to see when they have storytime.
The Field Museum just opened a new kid's section. I havent' been, so can't rate it. The Museum of Science and Industry is fantastic! There is an area called the Imagaination Factory (or something similar) that is great. They have a sectioned off area for non-walkers. If you check the museum websites, you can find when they have free or discount days.
Chicago has so much to offer children and their parents! And it doesn't have to be expensive!
Have fun!