You start off doing some basic movement stuff and whatnot, just do them as the game tells you until you finish the dialogue with Alonso.

1. The Captain: dialogue with some stat changes.
2. Henry (the Rum guy): Gives you a quest to fetch three bottles of rum (location will be explained below in the item section) and is a part of the payment quest. Owed 35 doubloons.
3. Alonso: Part of the payment quest. Owed 50 doubloons.
4. Boatswain: Some minor dialogue. His main function is the store.
5. Mathieu Pinchon (the Doubloon guy in the Wardroom): Gives you a quest to pay Henry, Alonso and Paul. Doubloon exchange. Do not touch the windlass before speaking to him.
6. Paul (the cannonball guy in the gun deck): Gives you a quest to fetch cannonballs from the cargo hold and is a part of the payment quest. Owed 25 doubloons.
7. Cargohold guy: Gives you a quest to find out the most important skill, which you can do so by choosing the lesson in lieu of monetary payment for the Rum fetching quest. Make sure to have a bottle of rum on you when you talk to him after finding the answer to the most important skill question, will give you an adder stone in exchange. If you don't have it on you, you will be locked out of it.

1. The chaser. The chaser/chasegun can be found near Alonso (where you started the prologue). Using it can give you +1 cannon skill. The interaction is unlocked after speaking to the captain. Make sure you select the option where you want to fire the gun.
2. The windlass. The windlass is the wooden postbox-looking thing before the entrance to the wardrooms. +1 navigation skill. Do not interact with it before speaking to Mathieu Pinchon the Doubloon guy. Doing so will result in him charging you extra for damaging the windlass.
3. The rum bottles. There are four bottles of rum. One in the cargohold that you can get for free any time. One is available for sale at the boatswain. One is the reward for the cannonball fetching quest by Paul. One is found in the captain's quarters, which you can unlock by passing a leadership check. You may need to buy the Monkey Fist amulet from the boatswain first.
4. The key to the captain's personal chest. The location is randomized. So far, I found it in three locations. One being the captain's desk, one being around the middle of the gun deck between guns, one being on a barrel on the main deck near where the Rum man is. You can press the tab key to toggle into a better angle to see find the key.

Above deck: One behind the captain. One next to the entrance to wardrooms. One next to the entrance to gun decks.
Wardroom: 3 chests. One near boatswain, one near the snoring gunner, one near Mathieu Pinchon.
Gun deck: 4 chests.
Cargo hold: 4 chests (1 of which is for the cannonballs)
Captain's Quarters: 3 containers. One wardrobe, two chests, one on either side of the room.

Loot them all and sell the useless junk to the boatswain get the treasure map.

Accountant gets a unique dialogue choice with the ship's purser during his mission to hand out doubloons. Accountant will receive 300 pesos instead of 150 for the reward and can exchange pesos at a better rate of 100 doubloon per peso (lower than the usual rate).

Duelist get a dialogue choice when the ship is boarded and you have to fight your first pirate. The normal options were 1. add weapon skills and decrease honor; 2. add leadership and increase honor. The duelist choice allows you to get weapon skills and increase honor.

Sharpshooter and athlete get a flavor dialogue choice each.

Once you have finished everything, go on to talk to the captain and tell him you are ready. If you looted his hat before hand (as you should), do not wear it when you talk to him, otherwise he will notice.

You will go through another railroaded section of the prologue. Once you get out, you should be able to get the Prologue and Prologue+ achievements.