What are the tools and procedures of a re-grout task in shower room?
I have done this many times in the past.


Make a weak solution of bleach and warm water (this works best) or if you don't want to use anything as aggressive then make a paste out of Bi Carbonate of Soda and Vinegar. Apply either solution to the affected areas and leave to soak for half an hour. Take an old toothbrush and scrub until the mould has gone.
Remove the grout (if needed). This can be done manually with a knife or you can buy tools to do this. Here are some Grout Removers. In my experience they don't work as well as they should. If you only have these 2 small areas, then just use a knife but be very carefull not to crack the tiles.
After grout has been removed, the sections can be re-grouted.
The area where the wall meets the floor can be siliconed after you have re-grouted. I am not sure which country you are in but this is the bees knees in the UK - it won't mould over! Dow Corning Silicone


Tips for applying Silicone


Apply the silicone in the form of a thin bead along the required area.
Make a weak solution of washing up liquid and water in a spray bottle and lightly spray the newly siliconed area.
Run your finger along the silicone to smooth it down


Hope this helps!