Try the complex of bars behind ul. Nowy Świat 23, sometimes called Chinatown. Not just the ones through the first arch since the ones through the second arch tend to be better. Don't be put off if some of them seem closed: try a door and you might find plenty happening. Lots of young adults go, students too.
20+ years ago there might have been something called 'student clubs' (ie. suggesting that this is a place for students only and nobody else). Nowadays, however, there are no strict rules; whether you're a student or not you're in as long as you pay for play ;). The best way to find out what the current top-places for students are in your area is to talk to other students. You may also check out some sites that list 'student places in Poland' (but, again, these are mostly places for general public). For example: warszawa.dlastudenta.pl/lokale/ in Warsaw.