Walk over to corner of State and Madison and take the Milwaukee 56 bus heading west. Ask driver to call Hamlin Street and then just sit tight. You will go through some crummy streets but when you get to Hamlin (about 25 minutes) you'll be in an old Polish neighborhood and right at the door of The Red Apple where all-you-can-eat of a huge home-cooked Polish buffet will cost you around $10 depending on whether weekday or weekend, lunch or dinner. Very good food---Red Apple has won many awards for best cheap food in Chicago. When you leave the restaurant turn left and stroll for a couple of blocks to browse Polish bakeries, grocery stores, and the Avondale Liquor Store which sells Polish Cherry [liqueur]. Red Apple has a website.
Not any real selection "downtown" proper near where you are staying so I'd say that jbw's rec for Podhalanka is a good one for being relatively close via a Blue Line L trip. Should you be willing to consider adding a short bus leg to that L trip I'd also suggest Smak-Tak for consideration.
For assistance in mapping your trips via public transit while in Chicago hit the http://www.transitchicago.com/ website.
Bobaks Saugage Company
5275 S Archer Ave
Chicago, IL 60632
Walk 2 blocks west to State street and catch the 62 Archer Bus (24 hours) heading south.
Bobaks has an impressive buffet and an amazing adjoining store!! Also try Gilmarts on Archer as well, Kurowski's (mind boggling array of Kielbasa, you feel like you're in Warsaw at this place, nothing like it) LOVE IT!
For all of your public transit info, 24 hours;
RTA INFO: 312-836-1000