New main objective: Speak to the Greybeards
Head to the Dragonsreach exit and you will come across Lydia, who the Jarl has mentioned (screen above). The warrior will now join your team and you will be able to explore the world accompanied by her (if a given character is following you, you can give always change the party member or kick him out). It's also worth noticing that Lyda can be given orders and also treated as a porter, by giving her items from the inventory during the conversation. Exit the building once you're ready.
Open the world map to learn that the High Hrothgar is located south-east of Whiterun. The only road to the top leads start at Ivarstead, a village at the foot of the mountain (screen above) and that's where you will have to go now.
If you have already visited the village of Ivarstead (or a different, nearby location), it would be best to use fast travel. Otherwise you will have to head there on your own. In such case, I'd suggest using the road-signs that will inform you of the direction (screen above). You will come across the first sign right after passing through the Whiterun Stables and reaching a fork in the road.
At first you will need to head east, but don't hope that the path will be straight. Afterwards you will turn south-east and eventually west. It's important to always stick to the main road and keep an eye on the road-signs that you come across, thanks to which you won't get lost. You also have to keep in mind that you might get regularly attacked by wild animals and Bandits. If you happen to come across a particularly strong enemy, try to just walk pass him. Reaching Ivarstead should probably take you around 20-30 minutes (screen above) and by far the best way of reaching it is riding on a horse (either bought or stolen).
Cross the village, heading west. Eventually you will reach a small bridge over a river (screen above), on the other side of which there's a path leading to the High Hrothgar. After crossing it to the other side, examine the emblem with a text fragment. On your way to the top you will come across nine more emblems, though reading them is completely optional.
Head to the mountain top. Try to stick to the stairs, as otherwise you might fall down form quite high. On your way you will probably come across a couple pilgrims, though the more important information is that you will also have to eliminate a few groups of Wolves (screen above). Make sure to stay away from ledges during the fights and attack one wolf at a time.
In the further part of the climb, e.g. in the area with the blizzard, you will come across a much stronger monster in the form of a Frost Troll (screen above). As the monster is only capable of attacking at close range and additionally is very good at it, you should attack him from a distance. If it's not possible, you need to be very careful and regularly use your shield, if you have one. In order to increase your odds of winning, it's worth to use fire spells and weapons dealing such damage. Additionally you can use Lydia or another follower to draw the monster's attention away, but not enough for him to kill your ally.
After dealing with the Frost Troll, examine its body and continue climbing up. In the final part, you shouldn't come across any monster anymore. Soon enough you should reach the High Hrothgar (screen above). Use the main stairs and afterwards locate the gate leading inside the building.