It is helpful to first understand the source of the mildew in order to properly clean it and thwart its return. Mildew is pesky little creatures that thrive on moisture and microscopic particles of organic materials—soap, shampoo, body oil, skin particles, etc.
Once you have mildew, even if you are able to thoroughly clean the grout, you will always have to battle it. Mildew sends out millions of tiny spores (seeds) that can lie dormant for a long time. Once moisture and the food source are present, mildew pops to life and sends out more spores into the air.
In addition to the mildew, you may have some hard water stains (primarily calcium deposits) on the grout. You will find this out as you attempt to remove the mildew because the cleaning procedures for the two are different. Also, you may have to treat the dull tile differently than the glossy tile.
First, go to your hardware store and buy a good-quality scrub brush with stout plastic bristles. The bristles may feel very stiff, but they should not scratch the very hard tile surface. Only use abrasive cleaners, even mild ones, as a last resort. Repeated use of these can dull all the tiles.
On your way home, stop at the supermarket and pick up some standard chlorine bleach, white vinegar, rubber gloves, and rolls of paper towels. You might also check out some of the common tub and tile cleaners that indicate that they are effective for cleaning grout.
Almost all of the common ones are safe to glossy ceramic bathroom tile. The dull, softer tile may or may not be a different story. You should test the cleaners on small areas first, or contact the company and ask them about it.
A few of the common bathroom cleaners to consider include Mist Away by Reckitt & Colman, Comet, Tilex by Clorox, and Original Clean Shower by Church & Dwight.
Use one of the above cleaners and the scrub brush to remove as much of the mildew as possible as well as any scum on the tiles. If you are having problems cleaning the tiles, it is probably hard water deposits. These are as hard as a rock.
Soak some paper towels with white vinegar and lay them over the tile and grout. The wet towels will cling to the vertical wall tiles. It will likely take many hours to soften the deposits. Check the paper towels every two hours to make sure that they are still wet with vinegar.
Rinse the walls thoroughly to make sure that all the vinegar is off and dry them. Wear your rubber gloves and saturate more paper towels with chlorine bleach. Place these against the grout and let the bleach do its whitening work for hours. This will also kill all the mildew. Make sure not to mix any chlorine fumes with any other fumes, as that can be highly dangerous.
In order to minimize the future growth of mildew on the grout, take away its nutrients and moisture. Rinse the walls when you are done showering and then run a shower squeegee over them. It only takes about 30 seconds. Leave the shower curtain or door open to hasten the drying time.