You can only add games from product pages (such as this one), by clicking "Wishlist it" under the social media buttons.
To remove the game from the wishlist, either click the button again (it will now be labeled as "wishlisted"), or go to your wishlist page and click the X button next to the game.
To share your wishlist, just copy-paste the link visible on top of your wishlist page (for example: GOG.com/u/YOUR_NICKNAME/wishlist). By default, you can only share your wishlist with people from your Friends list, but you can change the setting to "Everyone", or hide your wishlist from others by choosing "Only me".
To disable wishlist alerts (periodic emails notifying you when a game from your wishlist goes on sale), turn them off in your Subscritption settings.