Using bleach can be a smelly process, but it is one of the best ways of killing mould and its spores. I'd suggest after a thorough cleaning for you to seal the grout with Davco 500ml Sanitized Tile And Grout Sealer. The mould is hard to remove because it grows deep into the pores on the cement-based grout, and no matter how much you scrub, it won't come out easily. The sealer fills these pores so the mould can't enter them.
You should also find this step-by-step guide useful: How to seal grout. The Protector Respirator Kit would be suitable to help prevent inhalation of fumes. I would encourage you to swap out the cartridges that come with it for Protector A-AUS / P2 Combined Cartridge Filters as they will provide better protection. However, the key is to have the room well ventilated, so open some windows and put a fan on in there to circulate air. These cartridges protect against fumes and are also suitable for protecting against dust, mists and odours. You should replace the filters if they become blocked, you experience difficulty breathing or you can taste or smell contaminants in the mask. You also need to abide by any expiry dates stamped on the cartridges.