Clogged drains? can't fix them?

This is an odd question but how the heck do you unclog a drain? I live in an apartment and the drains are small. My shower drain is already so bad that the tub is half full of water when I am done showering. My kitchen sink is starting to get bad too although I never dump anything down it. I tried taking a hanger to get the hair out but it's not working. I feel like it's embarrassing and a huge hassle trying to call the rental company. Not to mention it took them a month to fix my broken heat in the middle of the winter. Would it work if I bought a bottle of clr or something?