The moisture in your bathroom makes it a great environment for mould and mildew to invade. If you have hard water, you will have additional headaches of hard-water stains and soap scum on your shower. Commercially available products may work well to clean the surfaces of your shower and other parts of the bathroom, but cleaning tile grout can require added products and elbow grease.
Scrub the grout using your scrub brush and the spray bathroom or mild abrasive cleaner. Apply the cleaner and scrub away as much scum, dirt, mildew or mould as you can. You will need to rinse frequently so you can see how dirty the grout still is. Continue to reapply cleaner and scrub until you can't remove any more scum, dirt, mildew or mould.
Apply vinegar to hard-water deposits on your grout using your paper towels. Tear off paper towels individually and drench each one in vinegar. Wring the paper towels so they aren't dripping and unfold. Hang a paper towel over any tough stains that remain after step 1. The wet paper towel will stick to the grout and even hang vertically. The acid in the vinegar works to dissolve the hard-water stains.
Check the paper towels every hour to ensure they are still wet. Remove each towel and scrub the area underneath. If after you scrub there is still a stain on the grout, drench the paper towel again in vinegar, wring and unfold, and apply to the grout. Repeat this step every hour until the stain is gone.
Wash away the vinegar that is remaining. Use your spray cleaner or mild abrasive cleaner and rinse thoroughly.
Drench paper towels in non-chlorine bleach in the same way you did the vinegar. Put your gloves on before you touch the bleach and be sure the bleach does not touch any fabrics you don't want ruined (shower curtain, rugs or clothing). Apply the bleach-soaked paper towels to the areas of grout with remaining stains. According to Tim Carter, whose syndicated column Ask the Builder appears in the Washington Post, "It can take hours and possibly days to bleach out all of the mildew in the grout."
You can substitute oxygen bleach for the non-chlorine bleach. Oxygen bleach does not have the bothersome fumes of non-chlorine bleach and is fabric safe. Follow the same process above as with the non-chlorine bleach. However, it might take significantly longer for the stains to be removed, according to Carter.