On the Blue Line, it is 20-30 minutes from UIC.
The area directly around UIC is not cheap. The area is becoming more expensive all the time.
On the UIC website, there is a listing of off-campus apartments but you will need an UIC netid to access it. I live in SSR. It is relatively cheap and convenient. Also, a lot of pharmacy students live here. Off campus, there are several near by apartment buildings. There are also lots of apartments off of Taylor and Polk. Some people live in Pilsen or Bucktown/Wicker Park area, and both of those neighborhoods are easily accessible to UIC. You can try Near West Realty; they are a local realtor. The average cost in the UIC area for a 1-bedroom apartment is $750-$1000, and for a 2-bedroom is $1000-$1200. Whatever you pick, remember the Blue Line is splitting. If you live in Oak Park or the northwest side, you will get off at Medical Center and not Polk St. If you live in Pilsen, you will probably get off at Polk.