Shower drains can get very dirty without much effort at all. You can clean your drain yourself without calling a plumber and without using any harsh chemicals. You first need to clear out the shower and the workspace.
You then need to remove the grate that closes off the drain so that you can access the pipe and the interior of the drain. You will need a cleaner that you can slide into the drain and then plunge in and pull out to help remove anything in the drain that is present.
The issue is going to be what is in the inside of the drain that is harder to get. To make sure that the drain is fully clean, you need to put a solution of baking soda and hot water. You can then let the cleaner sit and then use a larger brush to help grab onto anything that is on the interior of the drain.
Liquid soap helps to make the brush go into the drain better and helps to dislodge the clogs and the debris that might be present in the drain and that might be making it difficult to drain the overall shower. If you need further instruction, contacting a local plumber is a great next step!