First try to minimize creating the stains. Wait until your antiperspirant is dry completely before you put on your shirt.

To get the stains out of your white shirts:
– For old stains: soak the shirt in white vinegar or use a prewash stain remover.
– For new stains: soak the shirt in ammonia.
Then wash the shirt in the hottest water possible for the fabric using an oxygenated bleach or a product containing enzymes.