How do I get new spells for Warlock pet?

As of patch 3.02, all demon abilities and spells are automatically be learned as pets gain levels. Demon Master trainers have been removed.

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.