Remove the strainer and try a "ZipIt". Buy at Lowes for less than $5.00.

I would remove the strainer and use a hand auger. If the strainer doesn't have screws, gently pull it up with a screwdriver.

Clearing the 'clog' will usually require several remedies.

First, is this a shower/bath combo that has an overflow drain and plug?

If so, you will need to unscrew and remove the plate to the overflow to remove the stopper which has likely become clogged and blocked with hair.

See below:

This overflow plate is usually located about 4" below the spout.

Once you remove the plate, you will also remove the linkage to the drain plug.

With the plug and plate removed, run water into the drain and see if the drain runs freely.

If it does, replace the drain linkage and plate and your problems are over.

If not, try unscrewing the metal sieve, located directly on the drain in the base of the tub/shower.

Sieves can be come blocked with hair and soap. removing and manually cleaning them can open poorly flowing drains.

If none of that helps, you may need to unscrew the overflow and drain pipes, and traps that serve the bath/shower and clean them.

They are held in place in most cases by 'slip joint connectors' which can often be removed by hand or 'channel-lock' type pliers.

Separate these slip-joint connect parts...clean them...then reassemble...

And your problems should end.

Here is a link that might be useful:

Hair and soap buildup are pretty common in shower drains.

Mechanical removal of the crud is usually the only way to get them clean.

Occasionally the hair and soap start to even smell.