New main objective: Talk to Arngeir
Before you can meet with the Greybeard, you will have to leave the Sky Haven Temple. Good news is that you don't need to follow the same path as before, instead look around the room with the big wall for a set of stairs leading to the upper level. Use any one of the doors (screen above), open the world map and return to High Hrothgar. There, head inside the monastery.
Arngeir can be found in one of the corridors in the western part of the building (screen above) and you of course need to initiate a conversation with him. Tell him that you learned of the Shout that can be used to defeat a dragon from the Alduin's Wall. The Greybeard will state that if the Shout was used in the past, it seems that it cannot be used to successfully slay a dragon and he won't even help you in obtaining it. After en intervention from Master Einarth, he will luckily change his mind. Arngeir apparently doesn't know the Words of Power necessary for the Dragonrend Shout, though the Greybeard Paarthurnax can help you. Obviously, you will now have to gain an audience with him.
New main objective: Learn the "Clear Skies" Shout
Follow Arngeir, eventually reaching the High Hrothgar Courtyard. There, head to the spot where the Greybeards will want to teach you the new Shout. Wait for three different signs to appear on the ground (screen above) and approach each of them, therefore learning the Word of Power - Sky, Clear Skies, Word of Power - Spring, Clear Skies and Word of Power - Summer, Clear Skies. Arngeir will afterwards help you understand those word.