In that case, you probably want to stay north. In Chicago, elementary schools are zoned by neighborhood, so you'll want to pay close attention to that, and probably pick a few neighborhoods that feel "right", then make sure that the school is ok.
You may want to check out the Lakewood Balmoral neighborhood. It's bordered by Foster (S), Broadway (N), Bryn Mawr (W) and Broadway (E). It feeds into Peirce Elementary, which is thought of as one of the up-and-coming elementary schools in the city and has a very good principal. It's a neighborhood built in the 10's and 20's as a streetcar suburb that was annexed in the late 20's. Big wide streets, tall trees and victorian houses.
But that's assuming that you're going the public school route and not Catholic or other private school. Which is a completely different conversation.
My wife showed me this website a while ago, and I thought it was interesting. We're sending our kids to Catholic school, so I was more focused there, but it might be a handy resource for you:
Chicago Public Elementary Schools