How did naval artillery (like cannons) change the specifics of warfare tactics in the 16th century?

So I would first off recommend you go out right now. Seriously stop reading this, and go get it!

Empires of the Sea: The Siege of Malta, the Battle of Lepanto, and the Contest for the Center of the World by Roger Crowley.

Seriously a FANTASTIC book about the largest naval clash in the era taking place in 1571 in the Gulf of Corinth. Almost 500 total ships, the vast majority of them Galleys, though the Venetians provided 6 of a new type of ship the Galleass to the Christian fleet, which were of immense impact as they were designed with cannons in mind and mounted several.

The Christian fleet is held to have on the order of about 1800 guns across the fleet to the 700 or so in the Ottoman fleet. In addition about 7k high quality Spanish infantry were with the fleet, armed with arabesques, while the Ottomans were armed mostly with composite bowman. The Ottomans were also lacking a good number of their professional Janissaries, as they had been bled over the course of both the Siege of Malta, and the occupation of Cyprus.

Generally regarded as the largest galley of its time, the flagship of the John of Austria, also known as Don Juan( the illegitimate half brother of Phillip II of Spain) carried 9 mounted guns into the battle and 400 soldiers.

The battle actually played out as one might expect a clash of galley fleets to begin with both fleets in line in 3 divisisions rowing towards each other. The Ottomans were actually able to out flank the Christians on both sides, an inflict heavy losses in the center, but were never able to break moral. The large Galleasses were able to essentially act as mobile strong points, working their way through the melee and lending concentrated fire support where needed. By the end the Ottomans had lost 200 ships, keeping only 1 they captured, and the commander of the Ottoman fleet Ali Pasha had his flagship boarded 3 times by Don Juan's Real and supporting ships from both sides and was eventually killed and beheaded.

Now while their advantage on cannons certainly was of great use to the Christians, so was a willingness to close and take their enemies by boarding, which would of course remain a viable naval strategy until armor and increasingly accurate long range fire would become the norm 300 years later.

The other major naval engagement to demonstrate the flip side is the Spanish Armada, 20 years later. For that I would recommend:

The Spanish Armada by Robert Hutchinson

The English captains were well aware of the Spanish strategy to close to close range, blast away with their heavy guns, and then take ships by boarding. In all honesty EVERY navy would do the same in some form or another, the only variables being how long you would wait to close, and if you would do so at any opportunity or only as part of a larger plan.

In particular at Gravelines the English fleet was able to hold the windward position and pour fire into the Spanish, preventing them from moving to board. The Spanish ships were congested and full of supplies which made their gunners slow to reload, further allowing the English to fire more broadsides than they received. And while only 5 ships were lost it damaged many more, killed large parts of the crews, and put a stop to any further attempts to embark the Spanish army.

So I would say the rise of naval gunpowder artillery had 2 major effects.

It allowed individual ships to multiply their fighting power, and have a greater impact on the battle. Previously a naval battle would most always end in boarding or ramming, the ranged weapons they had only being effective out to a few hundred yards at most, so this allowed ships to begin to stand off from each other and trust in their weight of fire to win.
It upped the importance positioning and who held the weather gauge. At Lepanto between oared vessels it did not matter much. But during the Armada between Galleons(which were in the style and rig virtually all naval vessels would derive from) the English were able to use positioning to their advantage to deny the Spanish their preferred method of fighting. By holding the weather gauge, in other words being upwind of the enemy, you can choose when and where to engage, and how close to allow the enemy. The leeward enemy did have the ability to flee, but that would just turn into a chase, and until the rise of steam power(and even into that age as the wind would also clear the smoke from ships guns and their gunners views) most captains and admirals would prefer to hold the weather gauge.