Black mold is not something you want in your home, ever. Yet, it does happen and so from time to time we need to know how to deal with it. It does pose a health threat especially for people with compromised health or immune systems. Mold is a bacteria and will continue to grow as long as it has a source. Stopping it as soon as you see it will save you a lot of work and expense.
Spray on a liberal amount of bathroom tile cleaner. Allow it to sit two or three minutes and then start scrubbing. You will want to scrub away the mold completely. Rinse it with a good amount of hot water.
Keep in mind that there will probably still be a stain in the grout in the location of the mold. Apply a solution of liquid bleach and water 1:4 to any areas where mold was removed as well as any areas around it where spores might be settled. Allow the bleach to whiten the grout.
Remove the source of water the mold was growing on. If your bathroom tends to stay damp during the day, open a window or turn on a simple circulating fan to dry all the surfaces. If you can keep the surfaces dry, you will not have a problem of mold.
Apply a grout sealer to areas where the grout allows water to be absorbed. You can test this by splashing water on it and seeing if it beads off or sinks into the grout. Heat the grout up with a hair dryer to remove any residual water that has been absorbed and then paint on the sealer.
Allow the sealer to dry completely, usually for 24 hours before using the bath or shower. Keep the room as warm and dry as possible to keep any new mold from growing.