Port Royale 3

Ships are obtained 2 different ways; by buying them from a large shipyard (located in towns where a governor or viceroy reside) or capturing them during naval combat. You need to check shipyards regularly as new ships will only appear for a short time before other nations buy them, but shipyards will have more on offer every couple of days. 

TIP: At any time you can highlight a town and press  to see how many ships that shipyard has on offer. Unfortunately you need to either have a building or ship in town to see what ships are on offer.

You need to possess all the different types of ships at once and in a single game for the achievement to unlock. You will find the majority at shipyards, but there are a few that can be difficult to obtain. The following is the list of ships, the price and where you can usually find them.

Pinnace – 10,000 – Used by pirates early in the game/Given as the starting ship in Free Play 
Sloop – 19,000 – Used by several merchant and pirate convoys
Brig – 27,000 – Starting ship in Trader campaign/ Used by pirates 
Barc – 36,000 – Used by several merchant and pirate convoys 
Pirate Barc – NA – Can only be obtained by capturing it from pirates, luckily it is fairly common
Corvette – 36,000 – Found at several shipyards 
Military Corvette – 100,000 – Found at shipyards in towns with a viceroy/ Used by military convoys
Fluyt – 40,000 – Found at several shipyards/ Used by merchant, military and pirate convoys
Frigate – 70,000 – Found at several shipyards/ Used by merchant, military and pirate convoys
Military Frigate – 120,000 – Found at several shipyards/ Used by military convoys 
Galleon – 120,000 – Found at several shipyards/ Used by military and pirate convoys 
Trade Fluyt – 50,000 – Found at shipyards in towns with a viceroy 
Caravel – 160,000 – Found at shipyards in towns with a viceroy
Carrack – 140,000 – Found at shipyards in towns with a viceroy/ Used by strong pirate convoys 
War Galleon – 180,000 – Found at several shipyards
Liner – 200,000 – Found at shipyards (I only ever encountered these in shipyards belonging to the British)

TIP: It is often difficult to distinguish what type of ship you captured after a naval battle but you can pull into port, enter the Harbor Master menu and click and hold the left thumbstick, then scroll across to the ship list to match the name to the icon.