Downtown is fun, and there is some sightseeing to be done for sure. The best pizza in the city is Pequod's, it is deep dish not Chicago style, but it is a cab ride it's on Webster and Clybourn. Brix is also very good and closer to Downtown. For concerts there could be shows at The Vic, Aragon Ballroom the Riv or the Park West. Head up Willis Tower and stand on the overhang, that is interesting. The museum is great obviously and the Aquarium is really cool IMO.
Get on the El, and explore the city a bit. You can get the Brown/Red/Purple/Blue lines from the loop. The great thing about Chicago are the neighborhoods. There is any kind of restaurant you want in the city. Checkout Yelp and metromix.com to hone in on what cuisine you want. If you are hipsterish take the blue line up to Damen and hang around North/Milwaukee/Damen. If you want to ogle cute Lincoln Park Trixies take the red up to Clybourn and walk around Old Town/Lincoln Park.
Also, the UC is not in the greatest neighborhood so you may not get cabs after the games if it is like a Bulls game anyway. Figure out which Bus to take to and from and that will save you a bunch of time.
Have fun!