I gave my togetic a shiny stone to evolve into a togekiss but it wasn't able to learn aura sphere, sky attack, extreme speed or air slash.
To learn those moves you have to take it to the move tutor/relearner. Otherwise, it won't learn anymore moves than it already knows.
Togekiss only learns these moves at Level 1. I'm guessing your Togekiss is a higher level, therefore since you evolved it before Level 1, it's too late to learn those moves unless you go to the move relearner.