Coming from downtown [post code 60604] Harborside is probably the best overall bet.
Just played it last weekend.
Not sure about club rentals at any of the suggested places.
Another suggestion Marovitz. only 9 holes.
I have heard pace of play may be slow at times, but is close by
http://www.cpdgolf.com/Sydney-R-Marovitz-Golf-Course.html
Nearby driving range
http://www.cpdgolf.com/Diversey-Driving-Range-Golf-Facility.html
Two possible suggestions by Metra train, without a car this may be an idea.
http://www.ravisloecountryclub.com/
18 hole Donald Ross classic
http://www.dgparks.org/Golf/course-information-rates.htm
9 hole course, first course ever built in Chicago area.
Last suggestions if you are near the airport
White Pines, Oak Meadows, Maple Meadows.
http://www.dupagegolf.com/course/
http://www.whitepinesgolf.com/
Here's a good taxi service for getting around the suburbs or to and from the airport from the suburbs.
http://www.americantaxi.com/ATOnlineOrderWeb/main.htm
Getting to some of the fine courses in the outlying areas may be difficult with out a car or quite expensive by cab.
Feel free to PM me, If I have a day off, I'll show you around, maybe catch a round with you if you are interested.