I lived here off an on from 2000-2005. Finally got to go back last week. I think too many people end up just sticking around downtown. If you really want to get to know the city, get around. Here’s some of my favorite spots:

Lincoln Park Zoo (one of the only free zoos in the country)
Diversey or Belmont Harbor (amazing views of both Lake Mich and    the downtown skyline)
 The Empty Bottle (great little bar/venue with local talent and big name bands also)
 Hancock Observatory (MUCH better views than Sears Tower)
 Club Foot (awesome little bar w/ a HUGE Pez dispenser collection, free Tetris, and an amazing jukebox) 
  The Old Town Pub (an ABSOLUTE must see bar...open until 4 Am weekdays and 5 Am weekends. Right across from Second City. Old haunt of everyone from Belushi to Farley. DON'T miss it)