Not sure if downtown schools are convenient, or if you're interested in private schools, but we LOVE the British School. my daughter started when she turned 3 just about a year ago. she already reads and writes, does addition and subtraction, takes french, works a computer... i can't tell you how much she's learned. and she loves it, they have so much fun. it's a new school, so there are only about 400 kids i think. but the administration is very open to parent involvement and responsive to concerns. best of luck!

Where are you moving from? How old are your children? Are you talking about living within the City limits (some jobs have residency requirements) or just near the city, in order to have a reasonable commute to a job downtown?
The "city," including suburban areas, stretches more than an hour's drive from north to south, east to west....and cost of apartments/availablility vary tremendously. The City of Chicago has a magnet school program, and many of them have good extracurricular programs...my nieces were involved in track at their high school...Here's a recent discussion thread where another poster asked about moving to Chicago, and the responses and resources listed there might be helpful to you: community.greatschools.net/q-and-a/169992/Hi-Everyone-We-are-relocating-from-the-UK-and-need-advice-on-neighbourhoods-and-schools-in-chicago-Anyone?cpage=1#