For the most part bathtub and shower drains get clogged from human hair. The hair often gets stuck in the drain at the top or flows through and gets stuck in the p-trap below.
You will need a plunger, rag, needle nose pliers, multi head screw driver, Allen wrench, hacksaw and maybe a small snake. Do not use any type of acid or other drain cleaner.
If your bathtub is older then you may have an old drum trap instead of a p-trap. There is a drain at the bottom of some of these, but for the most part even if this is accessible the drain plug is normally rusted in place. And unlike the P-trap under your sink there is no union on a p-trap under the bathtub.
Try to source the clog before you get started. For instance if you run water into the bathtub and the tub starts to fill right away then you know the clog is close. If the water runs for quite awhile before the tub starts to fill then you know the clog is further down the pipe.
Some bathtubs have a chrome or brass plug that fits into the drain. You will need to remove this either with an Allen wrench or screwdriver depending on the type of drain plug you have. Some have the lever style that are controlled from a lever that comes out of the overflow pipe.
Before you try to plunge the drain you should try to pull out as much hair as you can with a your needle nose pliers. This will often resolve the problem. If not then you should attempt to plunge the drain next.
Before you start you should take off the over flow plate. You will need a common head or a Phillips head on your screwdriver to remove the one or two screws holding the plate in place. Be careful not to let the pipe behind the tub drop or you will have a tough time later on.
Next you will want to wet your rag and place inside of the overflow pipe. This will allow you to plunge the drain and force air into the p-trap rather then the air coming up overflow pipe.
Now place some water in the bottom of the tub so you can get a good seal on your drain. Now plunge the drain with your plunger while holding the wet rag in the overflow pipe. If you do not hold the rag the force of the water and air will push the rag out of the pipe. The best kind of plunger for this is the accordion style. Not only can you push air and water into the p-trap you can also pull water from the drain.
This might take some effort on your part. In fact you may have to plunge quite aggressively for 5 to 10 minutes to get the drain unclogged.
If after you have worked up a hard sweat you still cannot get the drain unclogged you will have to attempt to use the snake. If you have the type of plug that has the lever control you will have to attempt to remove this before snaking. In this case you may want to go beneath the floor where the p-trap is and cut out the pipe, and snake from there.
In some cases you cannot get the snake through the p-trap and you will have no choice but to go below the floor into the basement or the main floor of the house and access the p-trap. Unfortunately if the ceiling below is dry walled you will have to cut into the drywall to get access.
Once you do this you will either have brass, galvanized, ABS, or PVC pipe. Cut into the pipe and above the p-trap and just past the p-trap. This will give you access to the drain that you can now use a snake on. You will want a bucket to catch all the water from the p-trap and bathtub. If you have a full tub you might want to bail some of the water out before cutting into the line below. And if you USED ACID of any kind you should call a plumber.
Once the drain is clear you will need to assemble everything you have taken apart. Install a new p-trap if this is required. Using a Fernco which is a rubber clamp will help you out quite a bit.
Once the p-trap is in place you should test by filling the tub up and watching how fast the water flows out. If everything is good you can now place the overflow plate back on and repair any holes you may have made.