Obtaining the Required Skills or Items

You will need will need certain skills or items to complete all the steps. Most of these skills can be circumvented if you buy the right items, though you may need many for some of them. The skills listed under "Other skills" might be required, but might not, depending on the difficulty rating (stars) of the shipwreck map.

Fishing - Fishing is useful to obtain the shipwreck map pieces initially. Look at the Fishing page on the wiki to find out where to learn it and the requirements. You can buy fishing rods or reserve fishing nets instead if you do not wish to learn the skill, though you will probably need to fish hundreds of times as fishing up map pieces is rare. Alternatively, you can also very rarely find map pieces from combat or you can purchase them directly from other players for a couple hundred thousand ducats. (if you purchase the pieces, it is a gamble as you may not get back what you pay…) The collection skill at sea is also a good way to obtain map pieces.

Salvage - Salvage is needed to find and raise the shipwreck from the bottom of the sea. Look at the
Salvage page on the wiki to find out where to learn it and the requirements. You can buy salvage rope at the craftsman of larger ports if you do not wish to learn the skill, though you will need many as you will be using one each time to find out how close you are to the shipwreck. (If you do get the ropes, I recommend at least 20 of them as you may be using many to find it.)

Other skills:

Haul - Haul assists in towing the shipwreck back to the nearest port so you can search it by allowing you to keep your sails at maximum without needing to stop the ship for stabilization. (In other words, you don't have to be careful.) Look at the Haul page on the wiki to find out where to learn it and the requirements. You can buy haul rope at the craftsman of larger ports if you do not wish to learn the skill and this is recommended as you will only need one haul rope to haul it to port. (I recommend getting three or so, just in case the hauling gets interrupted somehow.) Haul (or haul ropes) is also useful for fleet-mates as when used, fleet-mates will add a cooperation speed bonus to the ship which is hauling the shipwreck when they activate the skill. (or use the rope)

Unlock - It does not appear needed for hoisting the ships to the surface (from my modest experiences so far) but is used while exploring wrecks, as some of the chests are locked/trapped. Some chest require the unlock skill, some don't. Higher difficulty rating wrecks are more likely to require this skill.

Archaeology - Some maps require this skill in order to salvage them and hoist to the surface. The requirement and needed rank appears under the difficulty rating (stars) on the completed-map window if it exists.

Finding and Raising the Shipwreck

First, you need to obtain a shipwreck map piece. I recommend fishing in your spare time, as rarely a fishing catch will be a map piece. You can also find them very rarely from combat with pirates and you can purchase them from players for a price. (You may spend more than you make if you purchase them, it is a gamble.)
Once you have a map piece, click on the small "Use Item" button in the upper-corner (either at sea or when walking around in port) and use the map piece. It will then be added to your shipwreck maps section. To view this section, click on the "Journal" button and then click on the "Shipwrecks" option. Here you can see the number of pieces you have as well as how many you need total to complete the shipwreck map. You can also see the total amount of shipwrecks you have raised next to "Cumulative Hoisted".
Next, repeat the previous steps in the procedure until you have gotten enough pieces registered in the shipwrecks section to have a complete shipwreck map. Before you set out to find the shipwreck, make sure you are fully stocked and fully repaired. Now look at the map and look at your map of the seas (or the maps on the wiki) and sail to the general sea zone which it is located. (If you have trouble finding it, use the maps here and make sure to try and match up unusual land shapes. Remember, if land leaves the map, don't assume what it looks like off the map as it could have any shape off the maps edge.)
Note: It is recommended that you bring fleetmates for the rest of this or at least have an aide, as either makes it much easier to find the shipwreck. Also, fleetmates present during treasure chest opening may get bonus treasure.
When you have arrived in the sea zone in which the shipwreck is located, the game will inform you by saying something like, "The ship you are targeting appears to have sank in these waters." When you get this message (or something like it, I may not be remembering it exactly) start using your salvage skill or salvage ropes. Each time you use the skill it will tell you if you are near the wreck. (When you are doing this with a fleet, each salvage use will tell you which fleet member is the closest to the wreck and that information can tell you which direction to travel in to get closer. This is an advantage over doing it alone.) Another way to accomplish this is to hire an Aide if you are high enough level, and assign them to lookout duty. The aide will inform you when you are close enough to the wreck by saying "look at the pieces of floating wood…" or something like that. At that point, break out the salvage ropes (it may take 2-3 tries to locate the wreck). Also, if you have an aide you can mouse-over the shipwreck map and you may find a feint red X. This X will be located in the general vicinity of the wreck, but will not be exact and you may notice it move about when you re-open the shipwreck map. When you have sailed very close to the location of the wreck it will change to a message saying the wreck is very near. When you finally hit the right spot with salvage, you will begin raising the shipwreck. You may receive some hull damage during this part, this is normal. (The speed you raise the wreck at as well as the damage you receive is affected by the total number of crew of your ship and also of your fleetmates. Thus, if you have more crew and/or more fleetmates you will raise the wreck faster and receive less damage.) Just wait and it will complete this part for you. (If you take too long raising the wreck or lose all your durability/crew the wreck will return to the bottom, but you can try to salvage it again and it will be in the exact same spot. In fact, its exact location should be displayed on the shipwreck map with a bright red X.)
Once the shipwreck is raised, it will appear as another ship on the water (called shipwreck) which is wrecked and thus has the white flag icon. Now you need to begin towing the shipwreck. (If you are in a fleet, this action will prevent you from following your fleetmates and vice versa.) To do this, just begin sailing to the nearest port and the shipwreck will follow automatically. If you do not use the haul skill (or a haul rope) you will need to watch the chat box as the game will inform you occasionally that you need to stop to stabilize or the towing will fail. (When using the haul skill/rope this doesn't occur, so that is recommended.) Once you are hauling the shipwreck, just head for the nearest port and dock there. Note that you will travel slower while hauling the shipwreck. (You will travel slower with the shipwreck in tow, but if any fleetmates use the haul skill or haul ropes, they can assist you and boost your speed.)
Note: If you take too long getting to port when hauling the wreck, the hauling will expire and the wreck will sink back to the ocean floor. However you can sail to where it just sank (red X on shipwreck map) and salvage it again.
When you dock at the port, you should see a new button while you are docked. (It will be a new button alongside the "Set Sail", "Provisions", "Employ Sailors", "Harbour", and etc.) Before you click that button, prepare yourself for on-foot exploration by purchasing land combat items such as medicine and darts or catapults. Also, make sure you have plenty of space in your cargo hold and in your inventory for the treasure you may find. (If you have any fleetmates who wish to join you, make sure they are sitting at the docks in port before you click the button to enter and they will join you. Fleetmates are recommended as they can receive bonus treasure and assist you with combat or unlocking. Also, fleetmates cannot loot the chests, so you don't have to worry about them stealing treasure.) Once you are ready for land adventure, click that new button at the docks and you will enter the shipwreck.
When inside the shipwreck, things are pretty straightforward. Search the wreck for any treasure chests you can find and beware of potential ambushes by buccaneers. When you are finished, exit the shipwreck by returning to the point you entered at and you will be returned to port with any treasure you obtained.
Note: if you die or leave the shipwreck with treasure remaining inside, you will be able to re-enter the shipwreck at anytime and at any port.
Afterwards, the shipwreck will be gone and you will have to start collecting more map pieces if you wish to search another one.