In Assassin's Creed: Odyssey, you can find small boats that are useful for reaching neighboring islands, however, in order to travel around the in-game world, you will need a large ship with a crew. This section describes when can you unlock Adrestia - your character's own ship.
You can unlock the ship after completing several first main quests. Acquiring it should not take longer than 1-2 hours. Your character receives his own ship before leaving Kephallonia - game's starting area.
Make sure to complete main quests in Kephallonia. After one of the dialogues with Elpenor, The Big Break mission will become available.
When completing The Big Break mission, you first have to meet the shipbuilder and deal with Cyclopes - a powerful mercenary intimidating local community. Upon reaching the indicated village, a fight with Cyclopes and his men will initiate. You have to emerge victorious in order to continue.
After winning the fight reach Barnabas and rescue him. Escort him to a safe location and then talk to him. You will learn that Barnabas is a captain of Adrestia, however, he is willing to give the command over to you.
Reach the Sami harbor where Adrestia is anchored You can now leave Kephallonia and end the first chapter of the game. From now on, you are free to travel around the world, improve your ship, recruit new crewmembers, complete naval quests and attack encountered ships.