July 2016 - Until the First Punic War, in 264 BC, the Romans had not been sailors, and had never had a navy. But when they had to fight the Carthaginians, who were descended from the Phoenicians and were great sailors, the Romans learned to build ships by copying a captured Carthaginian ship. Soon after that, in the 100s BC, the Romans conquered Phoenicia itself, and from then on Roman ship-building is really just a continuation of West Asian ship-building - the same old Phoenician sailors, only now they have become Romans.
By the time the Romans conquered Phoenicia, though, they controlled the whole Mediterranean Sea, so they really didn't need much of a navy anymore - just some patrol boats to keep down pirates, and merchant ships. Mostly the crews of these ships were Phoenicians, Greeks or Egyptians who came from sailing cultures. In the Late Empire, the Roman emperors needed all the tax money for foot soldiers, so they pretty much stopped supporting the navy.
Even though the Roman navy wasn't that important, when the Romans controlled the whole Mediterranean Sea, there were a lot of trade ships sailing around the Mediterranean. The people who designed these ships continued to try to improve them all through the time of the Roman Empire. The most important improvement was the gradual development of triangular sails, which first appear in the Late Republic, around 50 BC.
They may have developed from earlier Greek "brailed" sails, squeezed down on one side. These triangular sails gradually replaced the earlier square sails. We call these triangular sails "lateen" sails, because Latin speakers invented them. Lateen sails were a big advantage because with them you could tack into the wind. Using a lateen sail, you could sail faster than with a square sail, and the wind didn't need to be as strong to make the ship go.
Learn by doing: go out on a sailboat
I.C. Campbell, "The Lateen Sail in World History", Journal of World History 1995