Warlock - How do I get new spells for my pet?

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.

To place the new ability on your pet's action bar, open your spellbook by pressing “P”. Now click on the Demon tab at the bottom. This will show a list of all the abilities your demon knows. Now drag the ability icons onto your pet's action bar to use them.