If your drain is slow or has a stubborn clog, it's always easier (and much more pleasant) to try the path of least resistance. Start at the closest and most easily accessible point in the drain system and work your way down to the sewer pipes. You may get lucky and avoid the more smelly and dirty parts of the system.
Bringing up a hairball
You may have a ball of hair, soap scum, and other clogging agents just out of sight below the drain cup. Look across the spokes that cross the bottom of the drain cup for strands of hair that may be draped across them. These strands may be holding a suspended hairball.
If hairs are present. press a small piece of duct tape against them to lift them from the spokes of the drain cup. If you meet resistance, then a hairball is probably attached. You will need to pull the hairball to the openings between the spokes of the cup and cut away small pieces in order to squeeze the hairball through an opening.
Drain hairballs have a slimy, unpleasant texture, so grabbing a hairball with bare fingers is not advisable. Use a pair of pliers (needle nose type preferably) to force the hairball through the opening in the drain cup.
Even if no strands of hair are visible, you may still have a hairball lodged slightly out of visibility in your drain pipe. You can use a special type of hairball remover which resembles a plastic zip tie with numerous small hooks along its length.
When inserted into your drain and then retracted, these hooks snag loose clumps of hair or actual hairballs, pulling them close enough to the drain cup to pull them through or allow them to be cut into smaller pieces and then removed.
Sink drain traps
If no hairballs are found, you can try cleaning out the sink trap, which is the "J" shaped drain fitting just below your sink. Removing the sink trap requires an adjustable wrench to remove the large nuts on each end.
As you remove the last nut, hold the sink trap upright, or you will be exposed to foul water and sludge that sit in the sink trap to keep sewer gases from entering the home and to keep possible clogging agents from going farther into the drain system.
You may need an accessory to remove the sludge that lines the inner walls of the trap. A bottle brunch works nicely, but you can improvise with any instrument that will fit inside the curved trap.
When the trap is clean, wrap a few layers of teflon tape around the threads of the lower end of the trap before restoring the nuts.Teflon tape prevents leaks in threaded pipe connections.
Beyond the sink trap
If you still experience a problem after the sink trap has been cleaned, you will need to remove the trap once more to access the line to the sewer system. For this stage, you will need a drain auger (AKA drain snake), which is a round flexible metal cable with an auger tip. The auger resembles a spring with a pointed tip, and is used to snag clogging agents and pull them from the pipes.
The content of the drain lines beyond this point, raw sewage, is a thick black oily substance, foul smelling and capable of staining anything it touches, so thick gloves and a strong stomach are essential if you wish to proceed.
You will need to feed the auger tip into the drain pipe and twist it continuously as you proceed slowly along the drain line.If you feel resistance, you must then apply greater pressure as you twist and push the auger tip through the barrier.
When you feel resistance cease, you will then begin the process of retracting the cable and auger tip, hopefully with the clogging agent snagged by the tip. It would be wise to have a bucket or pail to wind the cable into as it is retracted. This will help to minimize contamination and staining by the sewage coating the cable.
If the clog still hasn't been removed, you will need to access the sewer line from the "Y" shaped clean-out fitting at the lowest point of the vertical main drain line in your home. You will need a pipe wrench to remove the plug that allows access to the clean-out portion of the fitting.
You will then insert the drain auger into the clean-out in the same manner as before, hoping to snag the offending clogging agent. If this fails, you will likely need to call in a professional, which might not be a bad idea earlier in the process (before you reach the raw sewage).