What was the transition like from the medieval oared galley to a sailing vessel? Does it parallel the same changes the armies faced from the switch to knights and men-at-arms to arquebus and pike? Do we see any internal changes within the navy? Ranks? Officer corps? Any other general changes, aside from the obvious technology like gunpowder?
Perhaps this post helps a bit, until one of our boat experts chimes in ;-).
To more directly answer your question, naval warfare developed quite a bit, in different parts of Europe, so I give a few illustrations here.
Spain in the Mediterranean
Spain's naval prowess evolved in nature over a rather long period in the development of naval power, and the development of the states that could afford to be a strong naval power.
Spain rose in strength (or more precisely the union of Castile & Leon, and Aragon, and Navarra) under Isabella and Ferdinand, at a time when Castile was the dominant power. In this period, Castile's major trading partners were the Nederlanders and the Italians. The discovery of the Americas and division of the world between Spain and Portugal meant that Spain had a new focus on the Atlantic, administered under the Council of Castile.
Following the reconquesta, expansion into North Africa also made sense due to growing need to import food items into Iberia, as Castile was increasingly focused on sheep-herding at the cost of agriculture. Under Charles V we saw a partnership with Genoa's galley navy in the region, where Genoa provided much of the naval assets and Spain provided the infantry to fight on the galleys and maintain garrisons (presidio) in the ports and towns they control.
In this period, the east coast (Mediterranean) of Spain, namely lands under Aragon, were in either upheaval or economic distress. So Spain's partnership with Genoa increased along with its strategic interest in the Mediterranean.
As Ottoman power in the Mediterranean grew, so did conflict with Spain and her allies. As we know, this culminated in the great Battle of Lepanto in 1571, where Spain's Philip II funded a large navy with Genoa, Venice, Papal State as partners. That victory was due to strong galley fleet that could maneuver under manpower even in the absence of favorable wind, carrying a large number of soldiers on board. The prevailing tactic was to fire once at close range as the galleys came close, then grapple, then send fearsome Spanish soldiers for boarding action. In fact, this is exactly the instruction given to Spanish soldiers. Those who had assisted with manning the guns were told to move to the upper deck after firing. The guns themselves are on older 2-wheel carriages, so reloading them was arduous if not outright impossible during combat.
The Battle of Lepanto itself was won through maneuvering and boarding action. The Ottoman flagship of Uluç Ali, the Sultana, was boarded by more than 3 Spanish galleys. Seven Ottoman galleys came from behind the Sultana and their soldiers came on board the Sultana to fight the Spanish. The Spanish commander de Bazan's own galley joined the melée along with a squadron of Spanish ships, turning the Sultana to a battlefield into which soldiers from both sides poured in to fight. In the end, Uluç Ali was killed and his head was displayed from a pike, crushing Ottoman morale.
English gunnery
As war with England loomed, you have keep in mind several factors:
Henry VIII Tudor was allied with Charles V of Spain, and his first wife was Catherine of Aragon, sister of Charles V. Henry had wanted to marry his first daughter Mary to Charles V, although Charles married a Portuguese bride with larger dowry. Instead, Mary was married to Philip II of Spain. This was part of Charles V's grand strategy of a northern orbit: Spain - England - Low Countries.
The period of Mary's rule of England saw Spanish naval knowledge being absorbed by English shipbuilders. Mary ordered the building of race-built galleons following Spanish design but with emphasis on gunnery. This includes more guns and better reloading mechanisms, in particular re-design of the gun carriages to use 4 wheels. So basically while the vessels were designed in Spanish style, the armament was done in English style.
Elizabeth, as we know now, was a Protestant and eventually led England away from Spain's orbit to become an outright adversary. She ordered further ships built following the design done in the era of Mary. These are the ships that led the battle against the Spanish Armada in 1588.
Speaking of which, the Spanish came into that battle first of all wanting to avoid confrontation with the English fleet. Their objective was to escort a large flotilla of transport vessels to the Low Countries where they could take the Army of Flanders across the strait to England, to conquer England. And this they nearly succeeded in doing, except for weather, English gunnery, and English persistence.
The two navies had very different goals. The Spanish would have much preferred to avoid the English navy. But they had to stay together as a fleet to protect the slow and vulnerable transports. If they had to fight, they preferred to fight on the decks, not from a distance. The English wanted to stop the invasion. They didn't want to fight a boarding action, they preferred ranged gunnery action.
The Mediterranean had taught the Spanish how to fight as a navy, culminating in their victory in Lepanto. But that style of fighting failed in the battle against England due to asymmetric objectives, advances in gunnery, differences between Mediterranean and Channel conditions, poor high-level strategy (most agree that the Spanish will likely have done better if they had either invaded England or Ireland straight from Spain itself, or have the Army of Flanders do their own crossing, rather than have to escort transporters literally around England first), weather, tactics, etc.
Other theaters of operation
I skipped over the Baltics, and various French attempts at building a navy, and of course the skirmishes between the Watergeuzens and the Dunkirkers. For those, I highly recommend perusing through the references below.
References