I've recently relocated to Chicago and am having fun trying out restaurants for the first time.

Heaven on Seven - pretty good spot with great gumbo. I like their cajun stir fry (although they could add more heat)

Bistro 110 - very good, already mentioned here

Sugar - excellent dessert spot. You won't regret it - although it is a little pricey

1492 - someone else mentioned Cafe Iberico, try this place for a little different atmosphere. The restaurant is located in an old house.

Mambo Grill - excellent barbecued salmon, great service, not too pricey.

Just a few that I would recommend. I'll have to follow sme of the other suggestions made here.