Around Chicago, population density decreases more gradually than most cities. There are a lot of suburbs - like Skokie or Evanston - that people elsewhere might expect to be part of the city. I'd suggest checking them out: they're both older neighborhoods, and Evanston is particularly good because the CTA connects to it.
That said, the areas in Chicago proper that The_Jazz mentioned are also quite family friendly I think. I've lived and worked in and around Lincoln Park and Lakeview for years now, and I think of them as a compromise between city living and suburban family-friendliness. I'm not quite as familiar with Lincoln Square, but I know it enough to know it's a good neighborhood and that the families I've dealt with there have seemed happy with it.
Figuring out where the decent schools are is one part, but once you've narrowed it down you should really take a weekend to explore the character of the neighborhoods as well.
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"Musicians are the carriers and communicators of spirit in the most immediate sense." - Kurt Elling
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