Old pets and mounts were right-click to use. New pets and mounts are consumable items that will teach you a spell.
You can access the Pets tab from the Character panel.
There are two tabs labeled 'Companions' and 'Mounts'. You can preview your pets and mounts here, as well as drag spells to your hotbars.
You can view your mounts through the interface. If you are mounted, the spell for the mount you are riding will be highlighted, even if you inspect another mount. This applies to pets, as well.