How do I clean discolored grout in my shower?

Start off easy and see how it goes: old toothbrush and "Comet with bleach" paste and add hydrogen peroxide, not water, to make the paste. Scrub it in, let it sit, and then scrub it some more, rinse.

Spray mold killers, like Zep (cheap and good) have a lot of off-gassing and will bleach anything, like your clothes and adjacent bath linens. Use extra ventilation and wear gloves. Once again, mechanical scrubbing helps get the soap scum off.

In this same category, are commercial grout cleaners, which are bleach based. You'll want one that addresses soap scum, as just bleach is typically not going to address all of this.

If you've very hard water, then use a product that has lime-removal or mineral-build-up removal components, like Calcium Lime Rust cleaner. Once again, the fumes will try to kill you, so ventilate extremely well and wear proper PPE.

Use the get toilet bowl cleaner with bleach.

First, wipe down the grout with hot water to remove surface dirt and grime. Next, spray a cleaning solution on the grout. You can mix your own by putting a 50-50 mix of vinegar and water into a spray bottle. Spray on the grout, let stand for five minutes and then scrub with a soft brush.

I like to use bleach and scrub it with a brush.