We lived there, out in the NW 'burbs, for about 9 months, but I am from north of Milwaukee originally, so its always felt a bit like "home".
Its a fantastic city with a great skyline, good green spaces, the LAKE obviously, always something going on, and its easily escapable on the weekends, or even for a day out. If you are childfree (like us) and don't have to worry about school systems, then you have a bit more latitude on neighborhoods, but really, anything on the north side will do, although I have heard near south side neighborhoods are up and coming. You may want to check out Evanston where there is a lovely little town center, has a great village feel, access to lake beaches, educated folk, great architecture, and El connections to center of the city. The city feel runs that far up along the lake, but without feeling like you are right in the middle (if you want more leafy green).
River North, spendy but trendy, and Lincoln Park areas too you could check out. Not sure your ages, but parts of the latter tends to concentrate new grads in certain areas.
Downsides: the traffic and the fact that a large portion of corporate jobs are in the 'burbs in every direction. AS much as we loved the city and wanted to live there, my job was in the suburbs and I didn't want to spend 1-1.5 hours each day, each way in serious bumper to bumper traffic out to Schaumburg. There is no such thing as a reverse commute in Chicago. The regional Metra trains are handy, but they only get you part way and then you need a car on the other end. Downtown also is absolutely dead at night, with places to eat/drink almost non-existant. Choosing your area is key because that is where you will do most of your socializing and living.