There are many interesting neighborhoods in Chicago. Many of them are focused on an ethnic group. And many neighborhoods have transition from one group to another.
Close into downtown there are several interesting neighborhoods. You can easily hit Greektown on the west side of downtown, then south to Taylor Street to Little Italy also known as University Village for the University of Illinois Campus, then down to Pilsen which was once Czech but is now Mexican, then over to Chinatown. This post explains how to make the trip.
Here are some of the neighborhoods.
Main St is Halstead
Little Italy aka University Village (U of I), Taylor St is main St
Pilsen (Was Czech, now Mexican) 1800 South west of Halstead
Chinatown - 22nd St is Main St - CTA train stop
Quick Neighborhood Visit
Greek Town, Little Italy, Pilsen (Mexican), and Chinatown
You can take CTA buses or Cabs