Talking to Serana unlocks the quest "Bloodline". Check your quest log:

Bloodline
While exploring Dimhollow Crypt, I freed a young woman from an ancient Sarcophagus. She asked me to guide her home, to a castle off the northern coast of Skyrim

Objectives:
Lead Serana to her home.

Make for the exit of Dimhollow Caverns. The gargoyles, who were previously statues, now transform into Gargoyle Brutes and attack you. Kill them.

Go through the hallway on the back, climb some stairs, and open a wooden door. Activate a lever there. Attack the Draugr Scourge and the Draugr Deathlord that come at you. There is a chest in this room so be sure to search thoroughly. Continue along the tunnels and up some more stairs. Reach a lit room with a torn banner hanging from the ceiling. It seems to be some sort of podium. There is a grate on the floor in the center with fire coming out. More undead Draugr attack you here, including a Draugr Death Overlord.

On the wall, there are some symbols. Approach them, to learn the dragon shout:

Word of Power - Stamina, Drain Vitality

Drain Vitality
(Stamina)
Coax both magical and mortal energies from your hapless opponent
Recharge: 30

There is another chest here. When you are done searching this room, leave through the door in the back, go up the stairs, and activate a chain to unlock the door. You will now exit Dimhollow Cavern.

If there are other quests you want to take care of, you can tell Serana to wait here. Although it is best to stay with her, to avoid any bugs where she disappears and cannot be found again. Also do not tell her that you think you should part ways, because there have been bug reports of not being able to get her to follow you again.

Check the map. The closest place to your destination is Lower Steepfall Burrow, so fast travel there. Make your way towards the water, and locate the big castle across. Find the Icewater Jetty. There is a small wooden boat near the pier. Step on the docks, then enter the boat and travel to Castle Volkihar across the lake. Once you have discovered Castle Volkihar, cross the big stone bridge, and approach the gates of Castle Volkihar.