I lived there for 11 years growing up we lived not even five minutes from Midway. I would look locally as most of the areas around Midway are just fine. Private schools are better if you want to live close but there are enough of them. The best part about living near the aiport is all the local food there. Even if you go to just look you have to try some of the Polish Delis there, their amazing. Good luck in your search Chicago has a lot to offer so im sure you wont have too much trouble.