Here's a list of things I think should be covered during your one month stay:
Wander Devon, see firstly the Devonathon thread on this forum. My favorite time is Friday afternoon when the Tel Aviv Jewish bakery is bustling and Good Morgan Fish is fully stocked but there is Indian chat and Georgia bread in the floor and secret (oral) menus of Russian pickles. It's a classic Chicago street.
I lived about 5 years in Uptown and every time I return, I wish I lived there some more. Get 3 curries and rice to go at the Thai Grocery. Take it to the Green Mill for a drink [OK, I've never actually done this, but I want too.

] but be warned that it can get quite crowded on weekend nights. The area nearby is one of the primary Asian neighborhoods in Chicago. Tank is very good for pho and other soup/noodles (I'm not as keen on their plates); Thai Avenue is an exceptional place. Convince them that you can take your food hot. Silver Seafood, about across the street from the Green Mill is a better than average Hong Kong style place. It's an interesting place, the side streets have some stellar achitecure. Find the old Essany Studio. It's gentrfying fast (very), but still rather mixed.
I think you'll like the Matchbox bar on Milwaukee/Chicago.
Check out one of the old (old) ice cream parlors. The only one remotely near downtown is Margies. It's good, but even better are Cupid, Cunnis or Gayety (the last two are in the South Suburbs).
Have an Italian beef at Al's.
Make your way to Cicero because, well you know, but once there, Freddy's is a great place for lunch. Klas has some Capone ties, it's not the best Czech food but still very worth a visit.
Eat Mexican food. If possible eat Mexican food on the Sunday Maxwell Street market. Regardless, look for places serving authentic regional items such as pozole, carnitas, mole de Teloapan, etc.
Visit the ex-Maxwell street 24 hour hot dog stands (more noted for their polish sausage and pork chop sammies).
Visit a farmer's market. I'm most partial to Oak Park's (where I live). It's Saturday AM. You can get your donut fix. While in Oak Park visit famous prairie school buildings including Unity Temple. If you need another Italian beef fix, go to Johnnies in nearby Elmwood Park.
If you cannot get a ticket to Wrigley Field, at least wander the environs. Try to snag a outta-parker during batting practice. Eat at Hamburger King.
Shop at a good Polish supermarket. Andy's Deli on North Milwaukee puts you quite near Superdawg, so that's a good thing. Avenue Mart on Belmont near Oak Park Avenue can get squeezed into your Oak Park visit.
Thai food. I've mentioned a bit already. Thai food in Chicago may be the
raison d'etre of LTHForum.com.
Think Yiddish, Dress British - Advice of Evil Ronnie to me.