If you have a bit of the black mold in your shower along the grout lines, a quick tip to make it go away is to use one of the Clorox Bleach Pens. You can find them in the laundry aisle, and because it squirts out the bleach gel in a small, controlled amount, it is easy to apply to the grout lines.
This is my shower in my own home; I applied the bleach pen gel to the black areas before I left for the day. When I got home, it was just a matter of rinsing it out of the tub. You don't want to just leave the bleach in there unrinsed, especially if your kids bathe in the tub.
If I am cleaning at a house and use the bleach pen, I usually don't have hours to let it sit. If I am really with it, I will go in at the beginning of the cleaning job and apply the bleach gel, and it can sit while I do the other cleaning. Otherwise, I apply it at the beginning of the bathroom cleaning and let it sit for the few minutes it takes to clean the other appliances. It can be enough to get most of the black out. I am usually good at making a mental note to let it sit longer the next time I come to clean at that particular house.
It isn't the end-all/be-all fix to cleaning the mold from a shower, but it is an easy, quick fixer upper. I usually do it once every other month, so it lasts for a while. It is just a quick, little tip to pass on to my readers!