Hi! We just recently moved out of a home with an old bathroom - the tub and shower were sealed incorrectly (they didnt allow the caulking to dry before turning on the water) and so black mold was growing both in and under all of the silicone sealing around the entire tub and shower area. No matter how hard or often I cleaned, it never LOOKED clean - I was disgusted every time I put my baby in the tub, even knowing that I had scrubbed that tub so hard one could eat off of it. I was never able to completely get rid of the mold - but I tried lots and lots of different cleaners and found something that works best. Did you know that the bottom of your bathtub is the most dirty, bacteria ridden area in your house next to your kitchen drain? The bottom of the tub also holds many bacteria that are truly harmful (things like antibiotic immune staph and strep, pre flesh eating bacteria and all sorts of really yucky things) so, it is SO important it be as clean as possible especially when you are bathing your baby in it.
1. Its a deep scrubbing that gets what is called a "biofilm" from the bottom of the tub: comet with bleach and a good rough scrub brush or sponge and some elbow grease will clean the tub part just fine. Put the comet down and scrub it in with water - then let it sit for a few minutes before rinsing.
2. Tilex works well for removing the black mold and mildew, but I prefer a simple capfull or two of straight bleach in a spray bottle. Spray the entire shower the the bleach solution - you can pick up the very edges of the silicone sealing and spray some bleach in there to try to kill the mold underneath it. Let it sit for 10-15 minutes. Scrub, then rinse. Repeat if necessary to get the desired whiteness. You might not get all the mold off, and that happens sometimes and you just have to deal with it.
After you do a really good deep cleaning, its just upkeep from there. We are in a new house now with a pristine remodeled shower and tub I scrub the tub with comet once every week and just use one of those scrubbing bubble shower spray cleaner things (they hang from the shower head) once a day after a shower. THe scrubbing bubbles shower sprayer really does keep the shower and basin disinfected and clean, and you dont even need to rinse it. (Im really good about having my bathroom and kitchen as clean as possible as I bathe and feed my child there).
Good luck, all it takes is one day and some scrubbing to get it really clean - and dont forget to open a window while you are cleaning so you dont suffocate from fumes!