My sink in my upstairs bathroom wasn't draining well. I did what I typically do and took my plunger to it when full of water. Since then, the water in the sink has just sat there and isn't even draining slowly. Prior to me doing this it would drain fine unless you ran the water for a while.

Any suggestions? And is using a plunger a bad thing on a sink?

You have a blockage in the horizontal drain line in the wall.
The solution would be to remove the "J" bend from the trap and send a snake up in to the wall.<(see image).
The good news is that once past a 90 that's about 8 inches in you won't have to put out over 5 or 6 feet of snake. Flush the line with a tub full od hot water. Good luck, Tom