While there are hundreds, if not thousands of family owned Taquerias and other Mexican restaurants all over the city and suburbs, you should start your search by focusing on Chicago's two primary Mexican neighborhoods.
The Pilsen neighborhood is centered around 18th Street and Blue Island. In addition to the many eating facilities you will find a plethora of Mexican owned shops and businesses. A little further west on 19th Street is the Museum of Mexican Fine Arts, recently designated by the federal government as the National
Museum of Mexican Fine Arts. The community has done an amazing job of building up a local park fieldhouse into a world class art museum.
Little Village branches out from 26th Street, west of California. You will find an environment similar to the Pilsen area with lots of small businesses and great ethnic atmosphere and flavor.
There are other, smaller Mexican enclaves, notably around Chicago and Ashland Avenues. Also north Clark street.
Even in my north suburban city of Des Plaines there is a heavy Mexican population with at least six different restaurants to feed them (and us), ranging from small taquerias to white table cloth places with full liquor service.
Let me know when you get to town and get settled. Once I know a specific neighborhood, I can make better recommendations.