The vast majority of slow shower drains are clogged with hair. I have had great success using a handy, inexpensive drain-brush which looks very much like a giant pipe-cleaner made of twisted wire and nylon bristles with a plastic handle at one end. This tool is specifically designed to clean out sink and shower drains, and is very easy to use:
Hold the handle in one hand and use your other hand to thread the tip of the tool down through the drain. Work slowly and carefully so you don't bend the wire core of the brush as you gently push the brush into the drain as far as it will go. When the brush has reached its limit, begin slowly pulling the brush up out of the drain. You will want to work very slowly, because the brush will be covered with slime and may be carrying a heavy load of hair that has caught in its bristles. As the hair emerges, try to grab it in order to avoid knocking it back into the drain. Pull the hair and the brush all the way out of the drain, then reinsert the brush. Repeat. The number of times you will need to repeat this operation is dependent upon how much hair has slithered down your drain since the last time it was cleaned.
If you cannot find one of these drain-cleaning brushes, look for a two-handled nylon-bristled brush that is designed to clean behind sink faucets; these two-handled brushes are, for some reason I cannot fathom, more common than the single-handled variety. Use wire cutters to remove one of the handles, and your modified tool will work just as well as an off-the-rack drain brush.