Belmont/Clark, Addison/Wrigley field area, Bryn Mawr, and downtown-State Street/ Michigan Avenue... Once in a while China Town is good, but only when i'm really craving Chinese Food. See previous person's post.
sidebar: people need to quit telling kids to go hang out at Belmont & Clark. it hasn't been like that for maybe 10 years. go out there at night, especially on the weekend and it's the closest you can get to Amsterdam in this city, which sounds great, but no "the punks" don't hang out there.
look up SouthKore Records on google and start in Pilsen instead. That's where the DIY punk scene is mainly based out of here. You go to one show and get flyers for the next one, that's how it works. those kids got shows going on every week, all year round. some bands you can look up on youtube from the REAL Chicago punk scene would be...Condenada, Chronic Seizure, No Slogan, Disrobe, Life Sentence, The Rat Bastards, Sin Orden, Pecadores, etc.
I say this so new kids don't come in here, go up to Belmont, think those the Belmont crews represent the scene, and then move out of Chicago in disgust.