You can miss the "feel" of an area when you just drive through in a car - a bike ride might be the way to go. The Chicago Cycling Club organizes bike tours through neighborhoods that are usually free and often include a meal at a local cafe or restaurant, a walk around and then a ride home.
On May 6, there is a Cinco de Mayo/Tour of Pilsen ride with a restaurant stop and walking tour. Ride home through Greek Town.
On May 27, a Bike tour of mosaic tile art thoroughout the city, ending with a BYOB picnic.
On June 3, a Southside park tour with a stop at Skyway Hotdogs.
On June 11, Explore West Chicago with a ride to Oak Park.
On June 18, a tour of Mies van der Rohe buildings with a guest docent talk.
And finally, on June 25, The Superdawg ride through the northwest neighborhoods of Albany, Jefferson, Gladstone, Norwood and Edison Park.