I'll second the
Billy Goat, under Michigan Avenue, as a classic Chicago hangout. If you're old enough to remember the old Cheezborger, Cheezborger sketches on Saturday Night Live, you'll get an idea of what you're in for.
I'd couple that with a trip on the red El line up to Lawrence, for the
Green Mill, the classic jazz hangout that Al Capone used to frequent. And before that, Charlie Chaplin was a regular, in the days when Chicago was the motion picture capital of the world, before those upstarts in Hollywood got so uppity.