Since some players are having difficulties with conquest mode, here's a simple way to win it. This is not an especially exciting way to win, but it's simple: no research, ship design, building defenses, espionage, etc.
Start with a small map on very easy mode. Save up money to build two HMS Unmatched, which are a default ship design.
Using the two HMS Unmatched and whatever ship you started with, start attacking towns, which are the ones with the small icons.
Your fleet should be big enough to induce them to just surrender. If not, retreat and build another HMS Unmatched.
Pillage each town you conquer and use the money to build more HMS Unmatcheds to grow your fleet. Conquer all the towns like that.
By the end of this process, you should have about a dozen Unmatcheds. You can now start conquering the cities. They won't surrender, but you should now have massively superior firepower. In combat, tell your ships to move close to the enemy ships and buildings and then just wait for them to be destroyed.
Whenever you conquer a city, pillage it, but wait until the pillaging is complete before moving your fleet to the next city, so it can restock its ammunition.
Keep doing that until you've conquered all cities and won.
Now you can go for a higher difficulty level and start looking into using your ships more effectively and creating better ship designs.