Chicago used to be known as a center for transportation because of its central location. Of course the dominance of the railroad has long since past but Chicago still cooks in the transportation department.
The RTA is the umbrella organization for all the local transit. In spite of severe cutbacks in the CTA, the Chicago area transit, the whole system is still pretty awesome.
I have a car and don't use it much but when I need to I am amazed how possible it is to get just about anywhere in the metro area on public transportation. Of course it takes a lot more time, but it is really inexpensive considering the alternatives.
This website is pretty decent for helping you plot your trip: