While playing through the story, you'll get various tasks from professors and upon completing them you'll get summoned to a class room or office and learn new spells. You have to perform the wandmotion once by pressing the button prompts for X, A, B or Y while following the spell motion.
You also have the option to learn the three unforgivable (dark green) curses, however refusing to do so does NOT lock you out of the achievement. Feel free to role-play your character how you want and choose the dialogue option to either learn the spell or not. The last one you need to learn is the transformation spell during winter season.