I'm a lifetime Chicago resident who works as a real estate attorney and a realtor in the City, so this question is right up my alley...
Living on a total monthly budget of $2,000 for a family of three will be challenging if you intend to live in a desirable part of the City with good public schools. I don't think it would necessarily be impossible, but I think a more attainable goal would be to try to find a 2 bedroom apartment and have your total housing budget cost about $2,000 per month. As others have said, most of Chicago's housing stock is very old and with that comes higher utility costs. I have lived in many apartments where gas (heat) costs upwards of $150/month in the winter.
I think a reasonable goal would be to find a two bedroom apartment in a great location for about $1,600 per month in and budget about $200 bills. If you think you all can squeeze into a 1 bedroom, then you can probably find a decent place in a great area for $1200-1400.
A car is definitely NOT a necessity in Chicago. The CTA (trains and busses) will take you anywhere. Chicago is also very bike friendly and I'd recommend getting a Divvy membership (it's a bike share type thing with bikes all over the city) for $75 per year. One nice thing about the Divvy compared to owning your own bike is that you don't have the risk of getting your bike stolen. Supplement the public transit with Uber and Zipcar and you should be good to go.
Now the question is where to live. As some others have said previously, my STRONG recommendation would be to live within walking distance to a brown line stop. Ideally I would try to live near a stop between Rockwell and downtown. The best, IMO in terms of family-friendliness (schools, safety, parks, other things to do) would be near the Western, Montrose, Irving, Addison, Paulina, Southport stops. Generally, housing is more desirable as you move south along the brown line so Southport will be more expensive than Western. With this in mind, target the neighborhoods of Lake View, North Center, Roscoe Village, and Lincoln Square. Lincoln park would be even better, but you're not going to be able to afford it.
As an alternative you could look along the blue line in the neighborhoods of Logan Square, Bucktown, and Wicker Park. These used to be the "hipster" areas and still have a lot of that vibe, so if you're into that, you'll love it over there. It's very safe, lots of great restaurants and trendy bars. I'm more of a Lake View/North Center guy personally, but would never talk anyone out of Bucktown area if that's your style.
The most mustachian choices would be Bridgeport, Pilsen, and Avondale. These are the most up and coming neighborhoods right now and will have much cheaper rents. The downside of course is that those areas are still developing, so the schools are not going to be stellar, and there could be some concern for safety. That said, I wouldn't be overly concerned with safety in any of these neighborhoods, but certainly you will find more rough characters in Pilsen and Avondale then Lincoln park.
Hope this information is useful. If you have any other questions or feel the need for some realtor help with your search, please feel free to PM me. Good luck!!