Wanted to chip in with a few comments. The close-in suburbs on the west side are also very nice -- thinking of Oak Park and River Forest here. Both have excellent public schools and a good few private elementary schools (RC, Lutheran, Montessori and Jewish preschools to mention just a few). River Forest is Illinois School District District 90; Oak Park is Illinois School District 97. The religious private schools go to age 14, followed by high school. The public elementary schools generally feed middle schools at age 11 or so, to 14, then HS. The more affluent suburban schools are excellent and I would have no worries about falling behind in any way except British and European history. Chicago public schools I would avoid like a plague, however.
Amenities in the Oak Park and River Forest area include Trader Joes and Whole Foods , nice public pools and great libraries (Oak Park especially), parks and playgrounds everywhere, with tennis courts, etc., proximity to the city without any of the drawbacks of city life (even though it's not all drive-bys in the city by any means, the suburbs are a bit nicer for children, a bit more child-scale).
The school day generally starts at 8, so a local school means you can sleep in a bit longer -- better not to choose a school too far away that requires a long commute for this reason, and also because you will meet people in your own neighbourhood if your children go to the neighbourhood school. Private school children in Chicago (city proper) come from all over the place and you don't get the chance to develop friendships with either parents or children when everyone is heading off in hundreds of different directions at the end of the day.
There are excellent opportunities for making friends with other young parents through school, preschool or sports. I really recommend you enroll your DCs in T-ball, soccer, park district sports and fun programmes, YMCA activities, swim team, etc., these are the best ways to meet people outside of the school community. In the more affluent suburbs, children do a huge amount of sports and other activities outside of school. If you choose a Catholic school your life will revolve around basketball, and homework.
Driving really is a must, even though there are trains and busses (trains and busses are really only in the close-in suburbs however). You really will need a car for shopping and getting around. The L goes out to both Oak Park and River Forest (a bit like the Tube but overground for the most part) -- Blue Line to Oak Park and the Green Line terminus is a hop and a step from River Forest, a matter of about 100 metres. The Metra (Geneva line) goes through both.
Chicago people can be a lot more reserved and 'cool' than New Yorkers or Bostonians hard to put a finger on it. If your DH is Irish, you'll have no problems being accepted and welcomed it's a little like Ireland West, and also very Poland West. Alumni clubs are very active and the people can be very, very helpful in getting you settled socially.
It's a sports-mad city, but there are lots of cultural attractions too, like the Symphony, Lyric Opera, theatre (a must), museums, and good shopping and nice restaurants. It really has something for everyone. The weather you just have to deal with it. Winter is long and snowy, and cold; summer makes up for it. There are nice vacation spots not too far away lots of people go to Wisconsin (Door County especially) or to Michigan for nice, simple lakeside relaxation. You can spend as much or as little as you want and have a good time.