After completing the quest "The Bloodstone Chalice", Garan tells you to go and speak to Harkon. This starts the next quest. Open your quest log:

When I returned to castle Volkihar, I was informed that Lord Harkon was waiting to speak to me.

Speak to Harkon

Talk to Harkon and ask him if he wanted to speak to you. He says that when he told you that he was grateful for his daughter's safe return, he told the truth, but he did not tell you everything. Tell him you suspected as much. He says that vampires are powerful, but have limits. Our great enemy is the sun. Until recently, it's an enemy we've had no way to fight. For centuries Harkon searched for an answer to this problem. He found an old prophecy written by a Moth Priest, those scholars who read the Elder Scrolls. The prophecy tells of a time in which vampires will gain power over the sun, and will no longer fear its tyranny. Harkon believes the secret to unraveling that prophecy is written in Serana's Elder Scroll. Harkon has ordered the court to assemble. He has a new task for all of us to carry out, and that includes you. He instructs you to come with him and hear is proclamation.

You now need to listen to Harkon's speech. Go to the audience chamber.

Lord Haron's Speech
Scions of the night! Hear m words! The prophesised time is at last upon us. Soon we will claim dominion over the sun itself, and forge a new realm of eternal darkness. Now that I have reclaimed one of my  Elder Scrolls, we must find a moth priest to read it. I have spread false rumors about the discovery of an Elder Scroll in Skyrim to lure a Moth Priest here. Now it is time to see if those efforts have borne fruit. Go forth, and search the land for rumors of a Moth Priest within our borders. Look to the cities. Speak to innkeeprs, carriage drivers, anyone who would meet a traveler. Go now, and carry out this task. This is my command!

Your next objectives are:
Locate a Moth Priest
(Optional) Ask carriage drivers about the Moth Priest
(Optional) Ask innkeepers in cities about the Moth Priest