For starters, go to Billing which you'll find under your workspace name in the top left corner:

Then, click View invoices and manage billing details. This opens Stripe's secure billing page in your browser.
Under Payment method, click Add payment method and enter your new card details. Your new card becomes the default automatically, and the card marked Default is the one we charge. You can remove a card you no longer use with the X next to it.

Note: Only workspace admins can access the billing page and change payment details.