There are two somewhat different things, slow drains and clogged drains.

With a slow drain, the idea is to let it drain overnight, then turn the water on and attempt to tell where the clog is by how fast the water starts backing up.  If it’s almost immediately, then that’s usually the trap right under the sink.  If it takes a wee bit longer than that, it might be the juncture in the wall.  If it takes much longer than that, then we’re talkin’ serious business and it’s probably time to call Roto-Rooter.

If it’s flat-out clogged, and it’s the only drain in the place to have a problem, chances are it’s the trap.  Grab the Channel Locks, slip a bucket under the trap and yank it off.

Plungers are also a useful tool, although most people don’t understand how they work.  It isn’t the plunger doing the work by pressing down; it’s the yanking up, pulling the water back up the pipe a bit, then the sudden release as the plunger lets go and the water drops back down the pipe and (hopefully) pushes things through.  In other words, it’s the weight of the water doing the work, not the plunger.  Plungers work fine in sinks and bathtubs and it’s definitely worth a shot if you have a slow or clogged drain.