Three diverse things off the top of my head that are not in the traditional museum campus/downtown/Gold Coast/Lincoln Park area, but very feasible without a car.

1. The Baha'i Temple in Wilmette is easily accessible by CTA train. On the way back, you can stop in Evanston, tour the Northwestern University campus and check out a very vibrant "suburban" downtown.

2. Oak Park and Frank Lloyd Wright (although I'd take Metra to Oak Park because the Green Line goes through some iffy neighborhoods).

3. Pilsen and Little Village for some of Chicago's Latino culture.