I would suggest getting a good map of the city (paper or electronic) and locating each of the neighborhoods and/or restaurants you're considering visiting. That way, you can judge walking distances for yourself. Remember, most of Chicago is laid out on a grid in which major streets are at one-mile and half-mile intervals, and 800 house numbers are equal to a mile (although slightly less in the immediate downtown area). You can
this website to find the geographic limits of almost any Chicago neighborhood. If you're using public transportation to get around, the maps on the
CTA website are a good guide, and a way of finding what is and isn't near public transportation. In particular, the el (subway, portions of which are elevated) is a fast way to get around. Places that are close to el stops are more convenient than those that aren't.
I think you might have a better time limiting yourself to a small number of neighborhoods (no more than 2-3) and spending a lot of time just walking around. Better yet, decide on 3-4 places (restaurants shops museums etc) which you absolutely want to visit within those 2-3 neighborhoods.
As for the downtown areas - which I might loosely define as extending from the Gold Coast to the South Loop and also including the Mag Mile (North Michigan Avenue), Streeterville, River East, River North, the Loop, and the West Loop - there are plenty of foodie-worthy destinations within that area. A "walking tour" of that area might include restaurants for almost any kind of food you can think of, as well as food stores and other sites. For example, leaving aside the hundreds of mid-priced and upscale restaurants, within that area you can find most of our local cheap eats (Chicago-style deep-dish pizza, Italian beef, Chicago-style hot dogs), small plates and wine-oriented restaurants (Quartino, Cafe Iberico, Avec, Bin 36), breakfast-oriented restaurants (Bongo Room, Orange on Harrison, Lou Mitchell's, Kitsch'n River North), our premier gourmet food store (Fox & Obel), bakeries (Sarah's Pastry), an excellent artisan chocolatier (Canady), etc. So don't rule out the downtown areas either.