New main objective: Demonstrate your "Unrelenting Force" Shout
After getting inside the monastery, wait for Arngeir to approach you and a conversation will automatically start. The Greybeard will want to find out himself if you really dispose the power of the Shout. Pressing Z should be enough (screen above), of course if you haven't replaced the Unrelenting Force with a different Shout in the meantime. In such situation, you would have to press TAB, choose Magic and set the Unrelenting Force as active in Shouts.
New main objective: Speak to Arngeir
New main objective: Learn the Word of Power from Einarth
Initiate a new conversation with Arngeir, from who you will learn a bit more about being a Dragonborn. In the end you will have to need to express your desire to continue learning more Words of Power. Einarth will teach you the second word of the Unrelenting Force. Wait for Einarth to say the Word of Power and afterwards approach the sign on the ground (screen above). It will let you learn the Word of Power - Balance, Unrelenting Force.
New main objective: Demonstrate your "Unrelenting Force" Shout (3/3)
Arngeir will tell you that apart from learning the Word of Power, you also have to learn its meaning. In case of the previous word (the one at Bleak Falls Barrow), it became possible by absorbing the Dragon's soul. This time it will be easier, as Einarth will share his understanding with you. Arngeir will ask you to test the Unrelenting Force Shout once again. Stand by the tile from which you read Word of Power and wait for your target to appear. The main difference from the previous time is that now you have to hold down Z (screen above) and not only press it. Repeat this two more times, getting rid of successive targets.
New main objective: Learn the Word of Power from Borri
Hear out Arngeir, who will tell you to follow master Borri onto the High Hrothgar Courtyard. Use the stairs and afterwards open the big gate. After reaching the new area, Borri will decide to teach you a new Word of Power. Wait for the sign to appear on the ground and approach it (screen above), thanks to which you will learn the Word of Power - Whirlwind, Whirlwind Sprint. Just like before, you won't have to worry about having to kill a dragon to complete the process, as Borri will share his understanding of the word with you.
New main objective: Demonstrate your "Whirlwind Sprint" Shout
New main objective: Speak to Arngeir for further training
Approach the spot where Wulfgar will present you the effects of the newly gained Shout, letting you quickly rush a dozen meters in a straight line. Before you use the new Shout on your own, you will have to set it as active. As a result press TAB, choose Magic and choose Whirlwind Sprint in Shouts (you can also add it to your favourites). Wait for the gate in front of you to open and quickly press Z (screen above). If you have do everything properly, you should get to the other side of the gate before it closes.
You can now return to Arngeir and talk with him for the last time. The Greybeard will tell you of your last challenge, which is obtaining the Horn of Jurgen Windcaller. The artifact has been hidden inside a tomb in the temple of Ustengrav.
Main quest unlocked: Retrieve the Horn of Jurgen Windcaller