New main objective: Read the note
Approach the altar and take the mysterious note left where the Horn should be. Open the inventory, head to Books and reach the note (screen above) to learn that the person responsible for stealing the Horn wants to meet with you at the attic room of the Sleeping Giant Inn.
New main objective: Meet with whoever took the horn
I'd of course recommend using the occasion and taking a look around the altar to find a few valuable items. Afterwards head to the door behind the altar. After reaching a narrow corridor, take a look around for a chest with precious treasures and continue heading forward. Eventually you should reach a door leading to the main part of the temple of Ustengrav. Keep going along the only possible corridor and you should reach an interactive handle. Pulling it should unlock the passage (screen above). Note that therefore you have reached the already explored part of the tomb, so now just head west to leave the temple.
Open the world map and fast travel to the Riverwood which you have visited at the very beginning of the game. After reaching the destination, find the Sleeping Giant Inn (screen above) and head inside.
After getting inside the inn, approach Delphine and talk with her. Choose the first dialogue option (screen above), asking her for the attic room. Delphine will direct you to the room on the left. head there and wait for Delphine to join you and state that she knows that you're the Dragonborn. She will give you the Horn of Jurgen Windcaller and suggest that you should meet in a different room and talk about other matters. Before you attend the new quest, I'd suggest finishing the previous one by giving the Horn to the Greybeards.
Main quest unlocked: A Blade in the Dark
New main objective: Return the horn to Arngeir
Head out of the inn, open the world map and fast travel to High Hrothgar. Head inside the monastery and find Arngeir (screen above). The Greybeard will congratulate you on completing all the challenges.
New main objective: Learn the Word of Power from Wulfgar
New main objective: Receive the Greybeards' greeting
Follow Arngeir and after reaching the main room of the monastery, wait for a sign to appear on the floor (screen above). Approach it to learn the Word of Power - Push, Unrelenting Force. Wulfgar will share the understanding of this word with you, so you won't have to kill any Dragons. You also won't have to test it, especially that it's just an extension of the previously learned Shout. In the end, stand in the middle part of the room and wait for the Greybeards to complete the ritual.
Now I'd suggest completing an optional activity, i.e. returning to the temple of Ustengrav. Use the same shortcut through which you left the temple, reaching the Ustengrav Depths. Head to the room in which you found the note from Delphine and interact with the altar to place the Horn of Jurgen Windcaller (screen above) on it, therefore gaining a free Dragon soul. You can now leave Ustengrav for good and approach A Blade in the Dark.