Mold in a bathroom is about as common as mold in last-week's refrigerated leftovers. Most of the time this mold is Cladosporium and generally not as much of a concern as the more toxic molds.
Cladosporium is the most common type of the black molds found in a home. It is black because it produces a black pigment that protects it from ultraviolet light. While some people can become sensitized to it and develop asthma-like symptoms, for most of us it is just a cleaning hassle.
The first thing you should do is to clean the mold from the shower. I'd remove and replace the plastic shower curtain and use a 10% bleach and water solution, or a commercially available tub-and-tile bath cleaning product to remove as much of it as you can.
Next, avoid future development of black mold by looking for ways to reduce the humidity in your bathroom. The easiest way is to make sure you have a bath fan, and use it! The fan should be vented to the outside and not into an attic, a condition I found quite frequently in the 20 years I spent as a professional home inspector. Add a timer to the bath fan controls to make sure it runs past when you leave the bathroom as it often take 10 or 20 minutes for the humidity for fully clear the room and not further feed a potential black mold problem.
You might also consider repainting the bath with a paint that resists mold and mildew. Likewise for the caulk that might be around the tile. You should also choose a product that resists mildew.
Finally, to avoid black mold in the bath, it helps to dry off the shower walls after taking a shower. In my house, we keep a small squeegee in the shower. Just a few strokes clears most of the water away from the walls, which then dry much quicker and help us to avoid the formation of mold.