The shower in my apartment has never dained well, I am always standing in water. All the sudden my bathroom sink drains slow. Also, when the sink starts to fill the water backs up in the tub. I can't shower now and have to brush my teeth in the kitchen.What should I do?

when the sink starts to fill the water backs up in the tub. I can't shower now and have to brush my teeth in the kitchen.What should I do
You have a blockage downstream from the tub. The solution would be to remove the tub overflow [plate and stopper assembly and snake from there.(see images)
You'll hit the return bend of the trap about18" down but once aroumd that you only have to put out 15 ' more of snake. Good luck, Tom