Skyrim Walkthrough Part 6: The Way of the Voice

After defeating the dragon and absorbing its power in "Dragon Rising", the Greybeards learned of this and have summoned you to High Hrothgar to test you as to whether you are really a Dragonborn and train you in the ways of the Dragonborn if you are indeed one.

Open your journal to see your new main quest:
The Way of the Voice
The Greybeards have summoned me to their monastery of High Hrothgar on the slopes of the Throat of the World. They seem to have learned of the mysterious power that I gained when I killed the dragon outside Whiterun.

Objectives
Speak to the Greybeard

At this stage you also have a follower, your housecarl Lydia. You may take her along if you think you need help, but it is not required, you can certainly complete this quest on your own.

From Whiterun, travel east, going past the Honningbrew Meadery, White River Watch, around the mountains to the north,  turning south when you reach Valtheim Towers. During your journey you should see signs pointing to Ivarstead. That's where you want to go. Follow the road keeping an eye on the map and the signs. Cross another bridge and keep following the road south past some ruins. Cross the bridge where the big waterfall is. Keep going on the road which will wind south ahead. Take the fork west at the signpost. This is a long road, but is the safest route. If you are feeling overly adventurous, you could have instead of taking the road, cut through the mountains, but then you face steep obstacles as well as risk plunging to your death. The paved road is the safest route to Ivarstead.

Keep going, past the Snaplet Cave, do not cross the bridge but keep south in the fork.  Cross the next bridge to the south, then continue west following the signs. Turn north at the next fork and cross one last bridge which takes you into the town. If it is dark, you may want to check in at the Inn for the night. Otherwise, take the bridge out of town to the NW to begin your way up the 7,000 stairs. Klimmek is standing near the bridge, you can talk to him and get a minor quest to deliver supplies to the High Horthgar for him.

Start making your way up the 7000 stairs. You will encounter a hunter name Barknar who will warn you to keep an eye out for wolves. Keep going on the path, pass through Karita who is meditating near an emblem and will also warn you about the wolves. You will also encounter a frost troll along the path, so be prepared to fight it or any other wildlife you encounter.

When you (finally) arrive at High Hrothgar, enter through any of the doors. Talk to Arngeir. Tell him you are here to answer their summons. He will say that they will see if you are the gifted one. They want to see you using your gift.

The quest log will be updated:
"I spoke to Arngeir, one of the Greybeards, who believes that I am "Dragonborn". which means I am especially gifted in the use of the voice. They have offered to help me learn to use my new abilities"

Your objective now is to demonstrate your Unrelenting Force shout to the monks. Use the shout power in front of them. They will be impressed and welcome you to High Hothgar. Tell him you want to find out what it means to be a dragonborn. Tell them you are ready to learn.

Your objective now is to learn the Word of Power from Einarth. He will teach you Word of Power, Balance Unrelenting Force.

Next, you need to demonstrate your use of Unrelenting Force shout by striking the targets as they appear. Face the panel on the floor. Press the button to shout ("Z" on PC). The longer you hold it, the more words will be spoken. You have to hold the button a little before the target appears, then release the button as soon as it shows up.

Your next trial will be in the courtyard. Follow master Borri to the High Horthgar Courtyard. Learn the Word of Power from him by stepping on the glyphs he makes on the ground, then approaching him. Then, demonstrate the shout you just learned. To use the Whirlwind Sprint to cross the gate before it closes, Stand near master Borri,, make sure whirlwind sprint is selected as your active shout, then as soon as the gate opens press and hold the shout button and move towards the gate.

After you pass this test, speak to Arngeir for further training. He will give you your last trial to retrieve the Horn of Jurgen Windcaller from the Tomb of Ustengrav, their founder.

This completes the main quest "The Way of the Voice" and starts the next main story quest, "The Horn of Jurgen Windcaller".