This quest starts as soon as you return the Dragonstone to Farengar in Whiterun. Note that this quest will enable dragon attacks as random encounters for the rest of the game. While they'll mostly appear in the wilderness, they may sometimes be encountered in cities or settlements.
After you return the Dragonstone, and Irileth will appear and announce a dragon sighting near the Western Watchtower. Follow Irileth upstairs, and listen to the discussion. When the discussion is finished, you will receive the reward from the previous quest, a random armory item, and permission to purchase a house within Whiterun.
Western Watchtower[edit]
Irileth will depart and head to the western watchtower. If you do not have a ranged weapon or spell, you should obtain one before continuing.
Head to the watchtower, either by following Irileth or going to the marked location. Soon after you approach the watchtower, a guard will announce warnings that the dragon is about to return. The dragon, Mirmulnir, will approach from the mountains to the south.
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The dragon itself is weak if you properly prepared for him, although he does become more powerful at higher levels. You can soften him with arrows with help from the other combatants, and when he is grounded, can attacked with melee weapons. It is likely that this dragon will fall quickly without too much difficulty.
After Mirmulnir is defeated, you will absorb the dragon's soul. The guards talk about this, now claiming that you are the dragonborn of legend. As such, you will now have the Shout power which allows you to use the various words of power that you later find on word walls. You might only have one right now, but you can obtain additional words from other word walls and unlock those shouts by absorbing other dragon souls.
Be sure to search the dragon's body for valuables. If you have enough free inventory space, you can even get the dragon bones and dragon scales.
To complete this quest, return to Whiterun and speak with Jarl Balgruuf. On the way back, you will hear a loud call, which is discussed once you reach the Jarl. Your reward for completeing this quest is the title of Thane (which gives the Axe of Whiterun, and Lydia as a companion) but your next step is to climb the mountain to reach the Greybeards.