That is a loaded question. Right off the top of my head, I could name a few dozen ethnic places to visit, not considering specialty shops. Here are just a few.
Check out some of the Balkan Deli's and Bakeries. I highly recommend Zoran Lalich's for the best Bacon I have ever eaten. He makes some darn good sausages also.
Lalich Delicatessens
4208 W Lawrence Ave
Chicago, IL
(773) 545-3642
Sandy's is also another good Serbian Bakery/Deli in the area.
Sandy's
5857 West Lawrence Avenue
Chicago, IL 60630
(773) 794-1129
If you are on Devon, you should check out City Fresh at Kedzie and Devon. They have a great selection of Sausages, Cheeses, Wines and Baked Goods from the Balkans.
City Fresh Market
3201 W. Devon,
Chicago, IL 60659
Tel: 773-681-8600
La Unica would be a good stop also for Pan Latin goods
La Unica Food Mart
1515 W Devon Ave
Chicago, IL 60660
While you are there you could stop at Devon Market across the street for Balkan and Eastern European goods
Also in Rogers Park, is the Caribbean American Bakery for Meat pies, Bula cakes and spice buns
Caribbean American Bakery
1539 W Howard St
Chicago, IL 60626
(773) 761-0700
My favorite Mexican markets are the Supermercados Morelia. They sell a great array of prepared food and fresh made tortillas and other baked goods.
Morelia Supermarket
7300 N Western Ave,
Chicago, IL 60645
(773) 761-3291
Morelia Supermarket #2
4833 W Diversey Avenue
Chicago, IL 60639
The Oakton Market in Skokie is fun because of the sheer size of its produce section which carries all kinds of ethnic produce (like 6 kinds of eggplant). The bread selection is pretty impressive also.
Marketplace On Oakton
4817 Oakton St
Skokie, IL 60077
(847) 677-9330
While you are there, you should stop at Eurostyle for Russian sausages and other goods.(including a great vodka selection)
Eurostyle Sausage Co
4861 Oakton St
Skokie, IL 60077
(847) 329-1430