First and most important is to clear everything debouchage out under the sink. You will need room to move around when disconnecting the drain lines. The next thing is to place the bucket under the drain pipe to catch any water that comes out. The last thing you should do is get a rag or paper towel handy to clean the gunk that you will inevitably run in to.

When you look at the drain line coming from the bathroom sink, you will see it make a "U" shape at the bottom. This is called the pea trap. You should have a large plastic or metal nut on both or one side of the pea trap. Use your channel locks to loosen the nut and carefully remove the pea trap. You may get some water running out at this point but it shouldn't be more than what is trapped in the pipe. Inspect and clean any debris in the pea trap.

The stem pipe is the straight piece of pipe that comes directly out of the bottom of the sink. If you have a stopper in your bathroom sink you will need to remove this first. There should be a nut holding in the stopper ball. Loosen the nut and pull out the stem with the ball. Now pull out the stopper from the sink above and clean any debris. Now you can disconnect the stem pipe. Most stem pipes will have a threaded connection and shouldn't be too tight. Try and loosen first by hand. If its too tight gently use the channel locks to loosen the pipe. Bathroom sink plumbing is fragile, so try not to put too much force on them to avoid cracking or bending the pipes and stripping the threads.