I have a tiled shower in an older house, and it's been a struggle to keep the grout clear of mold and mildew. I'm planning on installing a bathroom vent to prevent future buildup, but in the meantime, what is a good approach for a good, deep clean?
To remove surface mold I use a 4:1 mixture of water:bleach. Spray on liberally and allow to sit for about 15 minutes. Rinse with water and wipe clean.
If you need to use a brush to scrape off heavy deposits, I recommend brushing away from your face or preferably wear an appropriate mask and goggles. The mold spores are probably more dangerous than the bleach at that point, but neither one is fun.
Commercial cleaners such as Tilex tend to do a reasonably good job as well.
Whichever way you go, make sure that you give yourself adequate ventilation as the fumes may be quite strong.
As you've already noted that you need to take remedial action to prevent future growth, I'll just echo BMitch's comment that you should investigate if the mold has spread into the walls as hidden growth can be a future problem.
If there's mold behind the tile, which is likely given the age of the house, then cleaning will only push it back into the grout and delay the problem until the grout decays. Your best option over time is to demolish the old shower, put in a proper waterproofing membrane that's designed to be tiled over, (e.g. kerdi), and re-tile.