Most bathroom drain slowdowns and blockages start at the popup assembly, not at the trap or in the other drain lines. The culprits are the pivot rod and the stopper, which collect bits of hair and "stuff" over time. This mass grows ever larger until it eventually slows down sink drainage or causes a full-blown blockage. Unfortunately, drain cleaners will not dislodge this stuff... they can make it even more sticky and stinky!
In many bathroom sinks, the stopper is not removable by simply pulling it up because it is linked to the pop-up mechanism. Partial disassembly of the popup mechanism may be necessary to remove the stuff.
First, a brief description of the popup assembly. The stopper in your sink moves up and down by means of a metal rod (called the horizontal rod or pivot rod) that extends into the drain pipe right beneath the sink. This rod pivots when the lift rod is pulled upwards, raising or lowering the stopper. (See a blow-up of a pop-up mechanism and more troubleshooting/installation information by clicking HERE.)
If it pulls right out, you are in luck. Look into the drain with a flashlight... you can often see the blockage a few inches within the drain. Sometimes you can use a long screwdriver to push the blockage through. Then simply flush the drain with a few gallons of very hot water.
However, if the stopper does not just pull out with a gentle tug (don't be overly aggressive), it is interlocked with the popup mechanism. You will have to unscrew the popup rod under the sink, which is holding it in place. If you look underneath the sink with a flashlight, you will see the linkage that operates the stopper. Unscrew the large pivot nut located on the rear of the popup assembly (you may need pliers or a wrench), and pull out the rod, which will free up the stopper. It is alright to bend the linkage slightly to do this... just don't bend it any more than necessary to release the popup. When you are done you can bend it back.
(A note... when disassembling the popup mechanism, don't lose any of the small washers! They must be reassembled in the original order and direction or you will have leakage around the rod and nut.)
As I said, you will probably find lots of gunk on the stopper and the metal rod that drain cleaners will not remove. In fact, they seem to make it even more stinky! Pull out or push through what you can from the drain and clean the popup.
You can also flush the drain with hot water while the pop-up assembly is disconnected. Caution: Hold a sponge or paper towel tightly over the hole where the pivot rod was removed or water will pour out under the sink!!
If the blockage is too thick, you might have to disassemble the trap to get it all out.. The trap is an "elbow" under the sink that holds a small amount of water all the time. This standing water acts to block sewer gas from seeping into your home through the drains.
After reassembly of the popup mechanism, apply some lubricant, such as plumbers grease, to the threads, seals and the pivot ball on the popup rod. It will improve the action of the popup and make the next disassembly easier.
Flush the pipes again with a few gallons of very hot water to be sure all matter has washed through to the larger drain pipes beyond the sink drain. Again... please don't forget to reinstall the nut and pivot rod under the sink first, or the water will drain inside the cabinet!
As a "prophylactic" against future blockages at the popup, try to get into the habit of filling and draining the sink with very hot water every week or so. Due to the proliferation of low-flow faucets, most bathroom sinks have only low volumes of water running through them. This can cause the collection of all sort of stuff in the pipes... especially if you have a long sloped drain pipe running from the sink. This boost to the flow may help!