I second the Signature Lounge, Museum of Science and Industry and Art Institute. I’m not the most educated art person, but I enjoyed seeing all of the incredibly important and well known peiced they have. Several times I thought to myself "Man, this looks much cooler than it did in my Art History textbook.
Even if you’re not a baseball fan, go see a Cub’s game, regular seat, not bleachers. It’s not too expensive, and the old stadium is just magical somehow. There’s talk that they might take it down to build a new stadium for Chicago’s Olympic bid, so you could be among the last to see it.
Go for deep dish pizza at Uno’s or Duo’s or something. I’m sure you can find threads on here vigorously debating the relative merits of different pizza places.
Figure out the L and ride it. Look out the window and marvel at how the neighborhoods change from block to block. It still fascinates me, after years of living here and commuting every day.
Take at least one taxi ride at night up Lake Shore Drive, at night. Tall lighted buildings on one side, unbroken lake view on the other - awesome.