First of all, spend some time reading the "secondcitycop" blog-site to acclimate yourself to Chicago's current crime problems. Secondly, remember that Chicago is a quite segregated city - both racially and socio-economically - more than most other American cities, and "college-educated salary-earning white-collar folks" seem to congregate on expensive North Side of Chicago, in neighborhoods extending from Lake Michigan to maybe a mile or two west of Chicago River. Residential real estate prices bear this out. Chicago is rapidly losing its "middle-class paradise" halo. It can be quite expensive to live here, particularly if you've school-age kids (public schools are big issue here too, aside from some great selective enrollment schools), want a "nice" neighborhood, and/or want "nice" neighborhood amenities like attractive restaurants, stores, and streetwalks.
Chicago is also quite car-orientated, even for folks living downtown or in immediately outlying neighborhoods. Bike-commuting is big, but bike-car accidents are frequent too. (Our 100-person firm's seen five employees in bike-car accident hospitalizations in 18 months.) You might want to keep that car, at least until you decide whether public transportation fits for you. (CTA subway trains, for instance, are quite scuzzy, not like NYC's.) It's also not fun to take public transportation or ride your bike in 0 degree weather; trains and buses are often negatively affected too.
Rents in nearby to downtown, Lincoln Park, Lakeview, or Bucktown can easily be $2000/month for a one-bedroom apartment. Cheap Lincoln Park apartments are often handed down by DePaul University students, and are often quite worn-out. You might like Edgewater or Andersonville as well, nice lakefront communities north of Lincoln Park and Lakeview, and slightly more affordable, but you're facing a 30-minute subway ride plus commute to-fro. If you go further north to Rogers Park, you're facing a 45-minute subway ride to downtown. Chicago's working-class neighborhoods aren't nearly as nice, or safe, and are likely further from subway, or on west-side or south-side, which is another whole issue. (Go read secondcitycop site.)
Anyways, do some serious homework before deciding, and don't make the move thinking you'll save money because you're eliminating the car. Best wishes.