Give the shower a thorough cleaning with your usual cleaning products. Mildew feeds on soap scum, as well as the residuals of bath products and body oils. Getting rid of those gives the mildew spores nothing to feed on.
Wash your shower curtain and liner. Plastic curtains and liners can be wiped down with your regular cleaning products and then rinsed in the shower.
Mix a solution of half water and half vinegar and put it in a spray bottle. Vinegar kills mold and mildew and dissolves both soap scum and hard water stains.
Spray the vinegar solution on the mildew and scrub with a soft-bristled brush. Get into the grout with the brush and rinse frequently, spraying on the vinegar solution as needed.
Treat set-in mildew stains with a paste of oxygen bleach. Mix oxygen bleach powder with just enough water to create a paste and apply it directly to the grout. Let it sit for a few hours before cleaning it up.
Reapply the oxygen bleach paste, letting it sit each time, until your grout is white. Severe mildew staining can take several applications.