I think tax rates and methods of assessment vary from one municipality to another. Just for fun I check Realtor.Com to see what's available in my old home town of Berwyn,IL. I see huge variations in taxes on very similar homes. The City of Chicago is close to broke so I would be wary about buying anything there and I would get info on the tax structure of any location before making an offer to buy.
Where will you be working? There are several areas surrounding Chicago that are well-served by trains for reasonably pleasant commuting downtown. A broad generality that I think is still true is that property values on the north shore are very high compared to south and west suburbs. FWIW I have always loved the town of Riverside which was designed by Frederick Law Olmstead in the late 19th Century and most residents there can walk to the station to get a train into The Loop.
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