Try plunging if you don't have a snake. I'm having similar trouble (2nd floor bathroom). I plunged the drain rather hard and the
water started to go down albiet without the results I am seeking. I am hoping to get a snake from a relative soon as that advise seems wise. Remember you want to pull a bad clog out, not just through the pipe because it will only give you headaches later on. If the drain starts to smell foul then you have a really bad problem, that will be a cause to call the landlord or super or a plumber. Also I think it is imperitive that you keep the shower, tiles etc. clean and try to minimize the build up of mold and mildew. This stuff can get into the drain and form stubbon clogs too. If you do call a plumber tell him/her that you used drain-o or lye on the pipe so they are aware to put on gloves.