Upgrading the Jackdaw in Assassin's Creed 4: Black Flag is a long and sometimes arduous process. Not only do you need a lot of money, but in many cases you need a large amount of metal, something that is considered rare in the game world of Black Flag. Not only that, but to get some of the final upgrades, you need to find various items throughout the game. With all of this between you and a full-power Jackdaw, it's important to know what takes priority, and what will give you the best bang for your buck.
It doesn't matter how powerful your ship is if the armor surrounding the hull is less than stellar. This should be your first priority when it comes to upgrading the Jackdaw. The stronger your hull armor, the more attacks the Jackdaw can withstand before it sinks to the bottom of the ocean. It takes a considerable amount of money and metal to upgrade the hull armor, but you should get the intermediate hull upgrade before moving on to anything else, then come back to the reinforced upgrade as soon as you can.
The best strategy when it comes to taking down an enemy ship is to avoid their broadside cannons once you're within range. Likewise, you want to angle your attack so that the Jackdaw's broadside cannons do most of the work. The reason for this is because the broadside cannons on either side of the Jackdaw are the most powerful close range attack in the ship's arsenal.
When you upgrade the broadside cannons, you're not making them stronger. Instead, you're adding more cannons to both sides of the ship. More cannons means more shots fired, which leads to additional damage. This goes hand-in-hand with the heavy shot upgrade, which increases the power of the broadside cannons.
You get the first upgrade for free through the story mode in Black Flag. However, every additional upgrade costs a considerable amount in both money and metal. Get up to at least 13 cannons before you move on to other offensive upgrades. This should be your priority after hull armor upgrades.
The heavy shot is a powerful upgrade to the Jackdaw's broadside cannons. It essentially increases the firepower of the cannons, making them inflict significantly more damage with each upgrade. This should be your third priority as you go back and forth between upgrading the number of broadside cannons, and increasing the firepower with upgrades to heavy shot strength.
The first upgrade is fairly cheap at 900R, but it quickly jumps to 6000R for the next upgrade. The first upgrade should be done as soon as possible. However, do not upgrade a second time until you have at least reinforced hull armor and 13 broadside cannons. Anything less and you won't be getting the full potential out of the increased damage from the heavy shot. Having a lot of firepower doesn't mean much if you have a limited number of cannons, or if the Jackdaw can be sunk with minimal effort due to a lack of hull armor.
Hull armor, broadside cannons and the heavy shot are as close to mandatory upgrades as you're going to get. Beyond those three upgrades, everything else is option based on how you play. Mortars are very, very powerful weapons, but they're only useful at long range, and even then, only if you have good aim. If you like to take on other ships at close range with a surprise attack, the mortar shouldn't be a priority for you. However, if you like to get a head start on a battle, or play a hit and run game, put some time into upgrading mortar strength.
The round shot inflicts critical damage after a successful volley from the broadside cannons. The stronger your round shot, the more critical damage you can inflict. This is great supplemental damage to your broadside cannons. There's a reloading delay after each round of shots from the broadside cannons, which is just enough time to get some critical damage with the round shot. It's the perfect combination, and the proverbial one-two punch on an unsuspecting enemy ship.
The initial round shot strength upgrade is fairly cheap at 900R. You should get this upgrade whenever you can. It's unlikely you'll participate in many naval battles that don't allow you to use a round shot. After the initial upgrade, this becomes very much a secondary asset. You can inflict significant damage with the broadside cannons equipped with heavy shots, as well as from your mortar attacks. The critical damage from a round shot is effective, but not important when compared to the previously mentioned upgrades.
If you have a solid understanding of naval combat, there isn't much need for an attack that only shoots enemies behind the Jackdaw. Even if you're surrounded by a bevy of enemy ships, it's more efficient to make a sharp turn and use your broadside cannons than it is to use any of your rear attacks.
With that said, if you find yourself frequently getting chased, or running away from enemy ships, upgrade to the chase cannons (you start with the fire barrels, which can be replaced by chase cannons). Once you've upgraded, the chase cannons still shouldn't be a priority. Only use resources to upgrade them if you already have a good number of upgrades to the hull armor, broadside cannons and heavy shot strength.
Once you've crippled a ship enough to board her, you can use the swivel cannon to take out members of the crew, or target explosives on the deck of the enemy ship. In some cases, you can do enough damage with swivel shots that you won't even need to physically board the ship. With that in mind, the swivel shot strength is more important if you're not very good at sword-based combat. If you have trouble taking down enemy soldiers once you've boarded a ship, focus on upgrades to the swivel shot so you can limit your time taking down soldiers with Edward's swords.
Keep in mind the height of an opposing ship plays a big role in how effective the swivel cannon is, no matter how good of a shot you are. If the enemy ship is much higher than the Jackdaw, you won't have a good line of sight to take down the targets you need to kill. In addition, if you board from an awkward angle, you won't be able to turn the swivel cannon enough to target most of the crew. This is why swivel shot strength should not be a priority in most circumstances.
In addition to all of the upgrades previously mentioned, you can upgrade the chain shot and fire barrels, add your own visual touch to the Jackdaw, or invest in ancillary upgrades such as the dive bell or harpoons. Aside from the chain shot and fire barrels, these upgrades have minimal impact on your naval combat abilities.
The chain shot should be upgraded if you find yourself chasing down enemy ships. It only attacks from the front of the Jackdaw, so its uses are limited. In addition, even with a fully upgraded chain shot, it won't inflict as much damage as the broadside cannons or mortar shots.
The fire barrels are more of a preventative measure than anything else. They are in place to make it difficult for an enemy ship to follow the Jackdaw. They can be replaced with chase cannons, which are far more effective for mounting an offense, which makes the use of fire barrels very situational.