First ever visit to Chicago..
I was just in town over the weekend and had some of the best Mexican food at Nuevo Leon. Simple and fresh flavors in the heart of Pilsen. Pork carnitas and beef tongue were great and complimentary (4) salsas even better. Open every day 7 AM until midnight. Expect a line out the door many of the days. We went around noon on sunday. I also enjoy Avec and although it's out of the way try Hot Doug's for a variety of chicago style dogs. Good luck!
re: Pete Oldtown
I will make it a point to visit Nuevo Leon any time I'm in Chicago. Just to clarify, Avec and Hot Doug's were not visited on this particular trip. However I did find a great seaweed soup at a Chinese joint on Argyle. Paired it with duck tongues and was one of the tastiest hangover breakfasts I've had. it got me through the opening cubs game.
For Mexican try Frontera Grill or Topolobambo but you may be disappointed because of the hype for these venerated restaurants. I have loved both and been disappointed by both which is exasperating. I've had really good meals at Salpicon but, again, it can be uneven. For American I like Naha, One Sixty Blue, Sweets & Savories, Blackbird and North Pond but have very mixed views on Sepia and Agra Dolce and May Street Market. For Italian Spiaggia is always recommended but my last meal there was average considering the very high price point. Your mileage may vary because when they are at their best any of these restaurants can be truly excellent.
Well, I think the one "can't miss" specialty in Chicago is our Chicago-style deep-dish pizza. There are a lot of places serving great pizza here, and the chains (with the exception of non-downtown locations of Uno's) do quite a good job of maintaining consistent quality across many locations. Take your pick - double-crust "stuffed" pizza from Giordano's, Edwardo's, or Bacino's, or single-crust "pan" pizza from Lou Malnati's, Pizano's, Gino's East, or the original location of Uno's - you can't go wrong with any of these. You can phone your order ahead of time if you want to avoid waiting 30-45 minutes while seated for your pizza to bake; you can find their menus and locations on their websites.
We have some wonderful contemporary American, casual fine dining restaurants. IMHO the very best ones fairly near downtown are:
North Pond, in addition to having wonderful food, has an exquisite setting, too. It's right in the middle of Lincoln Park (the park, not the neighborhood), on the pond (the building used to be the place where people would rent ice skates in winter), with the city skyline as a backdrop at the opposite end of the pond. It's about three miles north of downtown, an easy cab or bus ride ( www.transitchicago.com ).
If I had to recommend just one of these places, it would be one sixtyblue for the food, and North Pond for the atmosphere, but there are lots of such excellent places; these are just the very best IMHO.
We also have some excellent mid-priced Italian restaurants, notably Cafe Spiaggia ( www.levyrestaurants.com ), Coco Pazzo ( www.cocopazzochicago.com ), Vivere ( www.vivere-chicago.com ), and Trattoria No. 10 ( www.trattoria10.com ). For more recommendations for Italian food, see www.chowhound.com/topics/360990
And, we have some excellent Mexican restaurants. And lest you think, "Oh, we have those at home", please realize that these feature very creative provincial Mexican cuisine. Take a look at the menus on their websites and you'll see what I mean. Ones near downtown include Frontera Grill and Topolobampo (both recommended above, www.rickbayless.com/restaurants ), Salpicon ( www.salpicon.com ), and Adobo Grill ( www.adobogrill.com ). For a summary of our best and most creative Mexican restaurants, see www.chowhound.com/topics/463572
Fox and Obel is worth a mention, too. It's an upscale gourmet grocery store, just a wonderful place to shop for prepared foods as well as basic ingredients. They also have a cafe in the rear that's open day for a meal or a snack. www.fox-obel.com
There are lots more places, particularly in outlying neighborhoods where you find more ethnic places (Thai, Indian, etc) but they involve some traveling from downtown (30-60 minutes via public transportation).
So those would be my first recommendations. You can find more detailed discussions about specific types of food here in these other topics:
Brunch and Breakfast: