www.chicagoreader.com and www.craigslist.com are the places to look for apartments, primarily.

I would look at Lakeview, Wrigleyville, Depaul, River North, Uptown, Lincoln Square, Lincoln Park, Roscoe Village, Wicker Park, Logan Square, and the Southport Corridor in particular given the info you provided. 60613, 60610, 60657, 60614, 60625, 60647 (maybe) in particular as far as zip codes go. I don't know the zipcodes for Wicker Park or Logan Square.

The Chicago Reader site has a map of all the zipcodes/neighborhoods on their site.

All of those are good for folks in your age group. You'll want to be near the L - brown line and/or red line is going to get you where you want to go, blue line works too.