What's the best way to clean grout?

First try cleanser with a toothbrush. Then wash that all very well Rinse well. Then try one of the commercial bleach sprays. Be careful, window open. It will bleach anything it touches - so old clothing.

I use Clorox Toilet Bowl cleaner on my silver grey and it works great. Have tried many other recommendations but found this one to work the best of any. Put a bit of the gel in the corner of four tiles and then brush along the grout lines with a toothbrush. Let sit for 10-15 minutes and then scrub again with the brush. Use a sponge or cloth dipped in water and vinegar to remove the soap residue and when that is all done, go over the whole floor again with warm water and 1/2 cup vinegar to remove any further residue. If you have a floor steamer then use that.

Yes, I use Tilex, let it sit for about 15 min. Use sift scrub with a scrub brush . Rinse well. When cleaned be sure to seal with a tile/grout cleaner so you won't have to work so hard next time.

if you have soap scum in addition to mold, then I would use something like lime away or calcium lime rust remover.

Both of those require a lot of ventilation, and rubber gloves and of course old clothes.

This video tutorial might help: https://youtu.be/DkASaY6JOHo