Cleaning your shower drain really depends upon what type of trap was
used when it was installed, and if the plumbing was installed
correctly in the first place.
There are two places clogs normally occur. The most obvious being in
the trap and/or waste line, not so obvious is an obstruction in the
vent, often at the fitting between waste and vent lines.
A plunger is most effective on P-Trap type traps, but offer very
little relief if a drum trap was used, and often what little relief
you do get is only temporary.
Assuming the drain used to flow freely and is now running slowly with
frequent clogging, if no foreign object is lodged in the trap, the
most likely culprit is simply a buildup of soap and hair in the trap
and/or waste lines.
Since your drain is only slow running, I would suggest pouring 3
gallons or more of near boiling water down the drain, pouring 1 gallon
then waiting about 2 minutes, pour the 2nd gallon and wait about 5
minutes, then pour the 3rd gallon of near boiling water. Then run
only the hot water tap for about 5 minutes to carry the waste all the
way to the street. A follow up of flushing the toilets will keep it
all moving and out of your waste system.