I used to live in Chicago and just came back from a trip there, so I have a few suggestions.
For food in Greektown, try Greek Islands or Pegasus. Many of the restaurants in Greektown (including Pegasus, I think) have rooftop decks, which are so nice when the weather is great.
In terms of other restaurant recommendations, I second the Lou Malnati's suggestion. I would also recommend a restaurant called Green Zebra. It is actually a mostly-vegetarian restaurant, with one or two meat/fish dishes. It is very popular, with most of their clients not being vegetarians. The food and flavors are just perfect and you really can't go wrong. If you are looking at the menu, know that the plates are deliberately a little smaller, so you would typically order three plates (unless you weren't very hungry).
Nicole's recommendation of Tru is great. It's pricier, but considered one of Chicago's best restaurants. Another recommendation, if you like Mexican food, would be Topolobambo or Frontera Grill. They are sister restaurants--right next door to each other. Topolobambo is a high-end Mexican eatery, while Frontera is more casual. I don't think you will go wrong with any of those suggestions. If you like Spanish tapas, I am a big fan of Emilio's.
As for shopping, Bucktown, Wicker Park and Lincoln Park are all good options. In Lincoln Park the boutiques tend to be more independent (although there is an Intermix and a Barney's Coop), but the prices can still be high. It varies, though. In Bucktown, they have a bunch of designer stores. Another place for designer stores is Oak Street, which is one block from Michigan Avenue's "Magnificent Mile."
I also have good hotel recommendations if you are interested!
