After the events that take place beneath Hyrule Castle, you awaken in the Room of Awakening.
Get the Sword
After the cutscene, check on the sword to take it. Then press Y to swing it at the vines in the doorway. Then go straight ahead and check on the glowing circle in the middle of the room.
Go up the stairs and through the door that opened. You will reach a platform with a green circle. Press the indicated button to jump, and you will jump into some water. Swim ahead, and you can press X to dash in the water.
Run up to the wall up ahead and you will climb it if you keep pressing forward on the control stick.
You will find two more green circle platforms. Jump down from each one into the water. After the second one, swim ahead and go to the glowing object to find a chest. Open it to get Archaic Legwear, which you should equip (press + to open the inventory, where you can equip it).
Go through the nearby door and you will find another ledge with a green circle. Jump from it and there will be a cutscene. After the cutscene, you can press the button indicated on screen to accelerate, allowing you to fall more quickly. You will land in the water. Swim to the stone steps nearby. You are now on the Great Sky Island.