The King is Dead - Long Live The King: Skellige
This quest begins automatically after completing the Destination: Skellige Story Quest. First thing you should do for this quest is actually optional, head over to Yennefer's house and equip the fancy attire that you find here. With this equipped head on over to the other quest marker on your compass/map to find Yennefer.
Note: During this quest you'll be able to romance Yennefer if you wish - I will tell you which dialogue options to select when we get to that point if you are interested.
Inside of the castle you'll sit at a table with Cerys and some others. Cerys will challenge you to a race which you can either accept or decline - beating her or racing her has no bearing on the rest of the quest. After some dialogue Yennefer will ask you to follow her and she'll fill you in on her true motives for being here - to pillage a laboratory!
Follow Yennefer and the quest will update a few times before you find yourself in a room with some haze and what appear to be taxidermied animals (pictured below). The animals will come alive and attack you, fight them off and you'll learn that Geralt is just hallucinating.
In the room after the taxidermied animals you'll have to use your Witcher Senses to find a mug of mead and a statue. There is a bunch more for you to investigate in here too but these two objects are what will advance the story forward. Place the mug of mead in the hand of the statue to open the "secret" door right next to it.
You will find the mask that Yennefer is looking for in the room that's through the "secret" door. Upon grabbing the mask you'll be immediately attacked by an Earth Elemental which is most likely going to be an extremely easy fight for you.
Earth Elemental Weaknesses: Elementa Oil, Dimeritium Bombs
Defeating the Earth Elemental will cause the room you're in to fill with poisonous gas. When the timed dialogue option appears (as shown in the screen shot below) you will want to select the "Damn, I wanna kiss you Yen" if you plan to romance her during this quest.
All that is left of this quest is a lot of dialogue - completion of this quest will give you the follow up, Echoes of the Past (Story Quest) as well as the quest Possession (Secondary Quest). This quest is the next story quest in Skellige for you to do.