We had to keep replacing the sealant round our bath and it is a job I detest too, so this time I have done it probably and it seems to be working. As you will see, I am slightly obsessed.
I read somewhere that the pink mould is caused by the pH of shampoos and soap, so rinsing the sealant off should help with that. I think the issue with black mould is ventilation.
It took me 2 evenings to scrape absolutely all the old sealant off with a stanley knife blade, then I put on a paste of oxygen bleach for an hour or so, cleaned the whole bathroom and left the room unused for 24 hours with the window open to let everything dry out.
If your bath has no movement when you fill it with water and get into it you can use epoxy grout as a sealant which works really well and doesn't seem to go mouldy.
If your bath does move do this:
First, you put a border of masking tape all around the bath at the width you want the sealant to be. Next, you use expensive silicone sealant (don't even bother with the cheap stuff) in a gun, and put a bead slightly bigger then the gap in the masking tape all the way around. Next (this bit is important to getting a neat finish) you spray the bead with a strong solution of washing up liquid. Next, you use a curved implement to smooth the bead all the way round the bath, wiping off the excess regularly with loads of washing up liquid soaked paper. Finally, peel off the masking tape and smooth down any edges that have been peeled up by the tape coming off.
If you have got any sealant where you don't want it, you can clean it off with very soapy water, if you leave it too late rubbing with talc works.
I hope we move before I have to do this again!