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After defeating the dragon Mirmulnir during the quest Dragon Rising, when you return to Jarl Balgruuf in Dragonsreach, he'll inform you that the Greybeards have summoned you. If you ask him who the Greybeards are, then he'll tell you that they can be found in High Hrothgar to the east, and that they might be able to teach you how to use your newfound shouting ability.
High Hrothgar is located at the peak of a mountain -- the "throat of the world." To get there, you'll first have to travel to the town of Ivarstead, which you'll find at the base of the mountain. The trip to Ivarstead will take you a while to complete, and you might want to stop off at any lairs or bandit camps that you spot so you can build up your character and your equipment. If you get lost along the way, just look for road signs. Each sign will have a pointer for Ivarstead.
On the northern side of Ivarstead, you'll find a bridge leading to the "7,000 Steps," which is what the locals call the climb to High Hrothgar. As you make your way up the steps, you'll meet two pilgrims, Barknar and Karita, who won't tell you a whole lot. You'll also run into an ice troll (not far past Karita), and possibly some other creatures, including bears, ice wraiths and regular trolls. Trolls are nasty, so make sure you have some healing potions with you, and also make sure your character is at least level 10.
Inside High Hrothgar, you'll meet Arngeir and the other Greybeards. Arngeir will ask you to demonstrate that you're really dragonborn, so you'll need to use your Unrelenting Force shout. If you haven't used a shout yet, then you'll need to go to the magic interface, select Shouts, select Unrelenting Force (the only shout currently available), and then shout it (by pressing Z on the PC). This will satisfy Arngeir, and he'll begin telling you a few things about being dragonborn and how shouts work.
Next up, Arngeir will ask you to learn a new word of power. To do this, all you'll need to do is stand in front of Einarth. Einarth will teach you "Balance, Unrelenting Force" (aka "Ro"), and he'll also give you the capability to shout it. Normally, you'll only learn words of power at special word walls found throughout Skyrim (typically in crypts and dragon roosts), and you'll make them available for shouting by spending dragon souls, which you'll receive each time a dragon dies near you.
Continuing your education, Arngeir will ask you to use your Unrelenting Force shout to hit some targets. The Greybeards will then cause some "spirits" to appear, and you'll need to hit three of them with both words from Unrelenting Force. To do this, just hold down the key you use for shouts. The longer you hold down the key, the more words you'll use.
Finally, you'll need to follow Bolli out into the courtyard (just to the north of High Hrothgar), where he'll teach you a word of power from the Whirlwind Sprint shout (the word you learn will depend on what words you've learned previously, if any). Then you'll need to demonstrate your new shout by using it to quickly run through a gate.
That will complete your introduction to shouts, but you won't be done yet. Arngeir will tell you that you have one last trial remaining -- to retrieve the horn of Jurgen Windcaller from his tomb in Ustengrav -- but this is covered in the next quest,
The Horn of Jurgen Windcaller