Herodotus has asked you to meet him at Thermopylai. After speaking with him a cut scene will initiate, at the end of which you will find yourself on your ship at the Hot Gates of Thermopylai.
Before you begin the long sea journey to Andros, Barbaras recommends that you upgrade your hull. You will find three pirate ships in the bay of Thermopylai, from which you can salvage materials that you can use to upgrade the hull of your ship. Sail out of the bay at full speed and begin the battle by ramming one of the pirate ships. You can then alternate between ramming, firing with archers and ordering spears to be thrown at the pirate crew depending on how close they are and how much time you have to prepare a ram. Once you have enough materials to upgrade the hull, you can be on your way to Andros.
Note: You will find a large quantity of loot at the bottom of the ocean as you sail to Andros. Use Ikaros to scout the surrounding area and identify the precise locations of the treasure. You can also continue to attack pirate ships along the way to gain more experience and loot that may go towards upgrading your ship.
Between the Cursed Land of Oedipous and the Dirfi Foothills you will encounter three particularly difficult pirate ships. Before you reach this point ensure that you have adequately upgraded your ship, especially the hull, so that you can withstand the onslaught of attacks which will come in quick succession. To defeat the pirates focus your ramming and fire on one at a time. Remember to brace if an incoming ram is inevitable to reduce received damage. Maintain mobility so that you can maximise ramming opportunities while avoiding theirs.
When you arrive at the western shore of Andros, dock your ship and use Ikaros to scout the area. You will then be able to locate the entrance to the mysterious structure. As you make your way towards it be cautious of the many bears which inhabit the island. It is best to shoot them from afar to avoid their devastating attacks. When you reach the other-worldly door, interact with it to initiate a cut scene. When the cut scene ends you will be able to explore the cave within. Walk to the end of the cave and interact with the glowing pillar to the left.
Interacting with the pillar transports you to the modern age. You awake as Layla to rest from the Animus, and you now have an opportunity to explore your surroundings. You can inspect various features of the room and use the laptop for a crash course in underlying Assassin's Creed history so far. When you are ready, return to the Animus and the Greek world.
When you return to Ancient Greece you will be notified that you can now upgrade your spear at the Ancient Forge. Killing members of the Cult of Kosmos is the method by which you earn upgrades. When you are ready, leave the cave. As you exit your sibling will fire a shot at you and beckon you to meet them above the cave. There you will enter a dialogue in which you can choose to fight them or attempt to remind them that they are your sibling first and foremost. If you choose the latter it will be easier to build a relationship later in the game, and to convince them that the Cult of Kosmos is a dead end. If you choose the former you will only alienate them further.
Note: SPOILER* -- Be mindful in particular of your interactions with family members. The more sympathy you show toward them, the easier it will be for you to convince them to join forces with you later in the game. This will not only provide a happier ending to the story, it will also allow you to recruit them as crew members of the Adrestia.
When you are done speaking with your sibling, return to Herodotus to inform him of what has transpired and complete the quest.