Answers

Almost any mildewicide will do the trick.  Grout sealants all seem to work but I would go specifically to a tile shop to see what they recommend.  I think most sealers will work but application and prep are key. 

Clorox foaming spray works great.  Just spray it on and leave it sit.  No need to rinse or scrub.  It may take several applications.  It is washed off when you use the shower the next time.  it will even remove mold behind the caulk after several applications.

This is a spot where cracks often develop. After cleaning with a a mildewcide, check carefully for a crack. If the grout is cracked, open the crack up a little wider with a grout saw (inexpensive at a hardware store) or steel blade and fill it with a color matched caulk or sanded sealer from the tile store. If the crack has moisture, let it dry well before caulking.

BLEACH IT! 

BOMB SQUAD ANSWER// HTH or better known as house hold bleach. As an Explosive Ordnance Disposal Operator House hold bleach can be your greatest weapon for biological items such as mold. Add a little dawn soap to stick to the surface.  Spray, allow to treat, and then scrub and rinse! 

Pres Gavigan Construction http://gaviganhomes.com/about-us/

Billy Gavigan 

You didn't mention if the grout was colored. Be very careful to NOT use any bleach based products on any colored grouts. Something that works good on all types of grout is bon-ami mixed with vinegar. Dig out any caulk before doing this. Let completely dry after cleaning before applying any sealer and finally caulking. We like and use Impregnator tile sealer.