Collect all of the pink crystals to unlock every ability.
The achievement will unlock once you have gained the last ability and completed the short training.
You can press RB (+LB for full map) to check how many crystals are still left in the area and needed for the next ability.
If you are missing only one or two in one specific area, it could be easier to move onto another area and unlock more abilities. Almost all of the crystals are easily available if you find a detour or a small opening close to it. Only a few of them require some tricks like gliding combos.
Tablets/the paintings are not required.