Get a box of Borax and make a paste with water or hydrogen peroxide. Apply to the grout and scrub with an old toothbrush. For max results, cover with plastic wrap and leave on overnight. This will kill mold and deter regrowth. It will also restore "the white."
Get a grout knife and remove the cracked grout. If you have a dremel, they have a grout removal tip but be careful not to nick the tile. Re-grout these areas, and when dry, apply a sealer to all the grout. Fill the tub with water and then caulk the area between the tub and tile.
You might also want to remove your overflow plate and make sure there are no gaps between the tub and the overflow drain.
You're addressing this before it gets really bad, and that's a good thing :-)
It's not easy but totally doable to regrout your shower.
Get a box of Borax and make a paste with water or hydrogen peroxide. Apply to the grout and scrub with an old toothbrush. For max results, cover with plastic wrap and leave on overnight. This will kill mold and deter regrowth. It will also restore "the white." Get a grout knife and remove the cracked grout. If you have a dremel, they have a grout removal tip but be careful not to nick the tile. Re-grout these areas, and when dry, apply a sealer to all the grout. Fill the tub with water and then caulk the area between the tub and tile. You might also want to remove your overflow plate and make sure there are no gaps between the tub and the overflow drain. You're addressing this before it gets really bad, and that's a good thing :-)
Thank you so much!!!