To progress the story you need to accomplish these 2 goals:

Talk to the Crew and Captain
Fire the Cannons

This is a pretty short segment of the adventure. After you take your Pirate Oath you can walk freely about the ship and talk to the crew. Take extra care to make friends with Sullivan. Talk to Bone, Cross and Lyle then take control of the cannons and do a little target practice.

When you have finished with everything there is to do up on deck, go see the captain in his quarters. The door is down on one side of the lower part of the ship over by Cross. After a brief conversation with the captain you will see a short movie and then you will arrive in Port Royal.