How do Warlocks gain new spells for their pets?

Near most warlock trainers, there is a demon trainer. If you speak to this NPC, you can buy items called grimoires which each contain a different spell. After buying the grimoire for the spell you want to teach, simply make sure that you have the appropriate type of pet summoned and right-click the grimoire.

Open your spellbook (press 'P' by default) and click on the pet tab at the bottom. This page will show you a list of all the spells your pet knows. You can then drag these abilities to your pet's actionbar.