There are a couple great options for cleaning your shower curtains, the first is for machine washing while the second is for hand washing.
For machine: Wash your shower curtain in warm water with vinegar or gentle laundry detergent. You can also use baking soda or a homemade solution of 1 cup vinegar and 1 tablespoon bleach per gallon of water. Hang the curtain to dry on the shower rod after removing it from the washing machine and shaking or squeezing excess water out of it.
For hand washing: You can easily clean your shower curtain with just a damp cloth and some baking soda. Next, scrub the entire curtain with the cloth. After scrubbing, rinse it off and then try to get any soap scum or hard water stains out of the shower area with a new damp cloth. If you’re still not entirely satisfied with the results, try using a second (clean) cloth to scrub all of the soap scum or hard water stains out one more time.
Here’s a great article to check out for more info.