Dan's of Central Florida offers some great tips in response to a similar question from last year:
As far as re-caulking, there are some mold-resistant caulks on the market. Look for specialty tile suppliers in your area.
A good quality scrub brush with a bleach and water solution will do the trick. Do the entire shower/tub area.
Then remove any older caulk by carefully scraping it away. Tile saws, sharp knife, scrapers etc. Then re-caulk using a good quality bathroom and kitchen caulk with Microban in it. This will help prevent new development of the mildew.
Once that is all done and the caulk is dry, about one day. Then using a good quality sealer, reseal the grout and tile.
The mold, mildew or what ever you want to call it is feeding on the soap, the dead skin cells and stuff we do not want to talk about that has become embedded into the surface or stuck to the surface of the tile has provided a food source for this to grow. By cleaning and sealing you will make it easier in the future to keep the shower base clean so there will be less gunk to support the growth you are seeing.
It is not a pleasant job to do, but if done properly cleaning will become easier for a few more years.