The Holy Order of Alta had always been a regional maritime power, with a relatively strong navy in the capital and many harbors that enabled sea trade, allowing the people to learn seafaring naturally. The presence of a competent navy was also essential to protect the island possessions of the country as Alta controlled the passage between the Mediterranean and the Black Sea, as well as from Asia to Europe. Apart from the countless folklore and stories of seafaring adventurers as well as the chapters from the Holy Book, the order itself had come to the lands of the Old Empire via sea and the people didn't forget their roots.
That's why when Aelius reformed the shipbuilding techniques and introduced new manufacturing principles, such as frame-first building instead of hull-first building or bending wood by the use of steam instead of cutting and fixing wood with nails and gluing, as well as the usage of the compass, to which people quickly adapted. The most difficult change was the introduction of advanced sails and new sailing methods that required a high degree of expertise and even further modification as Aelius himself only knew the principles and never experienced building or sailing itself. This resulted in postponing the introduction of a fully sailed ship until later, and more effort was put into making the xebec, a kind of mixed sailed and oared ship that was popular in the Eastern Mediterranean as well as among the Barbary pirates. Xebecs were not as fast as galleys in coastal waters as the presence of a sail meant a decrease in the number of oars and their effectiveness, however, the xebec was faster in open seas and in wind, as the sails allowed faster movement, and since xebec was larger than a galley, it had higher destructive power per vessel, plus its deck was higher than a galley, making it harder to board or shoot at from below and making it easier to defend from a xebec.
After the Holy Order of Alta declared war on the Kingdom of Pergamon, Aelius sent the majority of the fleet to conquer the coastal provinces and crush any enemy fleet they could see. The fleet was commanded by Admiral Niketas, an unusual half-dwarf seaman who had a pet leopard and would jump into the ships of the pirates with his leopard and make his pet leopard attack the enemies while he himself would throw heavy iron war darts and throwing axes in his early days. Due to his years of service as well as his willingness to follow him, the Crown Prince of Alta Aelius rewarded him with the newly established title of Admiral of a fleet, as well as giving him two leopards that he himself tamed and grew to the maximum size they could get. Niketas was put in charge of the largest ship in the fleet, which had 12 guns on each side.
The fleet quickly sailed down the Aegean coast and took control of the islands and the coastal cities that the Kingdom of Pergamon ruled. The majority of them surrendered without a fight, but those that prepared for a siege even after being told to surrender were bombarded until they surrendered, and had their garrisons taken prisoner and their lords sent to Pergamon where Aelius was. The castles and towns that didn't surrender even after a bombardment was then besieged depending on their proximity to the coast and were conquered. Admiral Niketas had around 2000 men ready for a siege or a confrontation on land, therefore he was told to occupy only the coastal towns, cities, and islands as going further inland would make the marines an easy target for the more experienced Almagovar warriors, who excelled at guerrilla warfare.
After taking control of the majority of the coastal towns, castles, and islands of Pergamon, Admiral Niketas set sail to the island of Chios, the island which Aelius got his mastic gum/ rubber alternative from. Chios had three mountains and many hills from which attacks could be launched, as well as well-defended walls. Unlike most islands of Pergamon, Chios had a strong Catalan minority, meaning they had many Almagovars, who were ever keen to take advantage of the terrain of their homeland to ambush and attack the invading army of Alta. However, the first line of defense of the island was the fleet of Pergamon docked in its harbor.
The fleet of Pergamon was both outnumbered and outclassed, however, their aim was not to engage the fleet of Alta head-on, but rather blockade the narrow passage between the continental land and the island of Chios on its Eastern side where the only natural harbor of Chios was located. The Captain of the forces of Pergamon thought that If the fleet of Alta decided to land on a different part of the island, then they would've had to go through rough terrain and were exposed to the attacks of the Almagovars at night, and they would've exposed their fleet to the high tides. On the other hand, he thought that if the fleet of Alta tried to attack head-on, then they would lose considerable amounts of troops and ships, making it possible for the conquered territories to rebel and push out the garrison of Alta's soldiers that were placed in strategic locations and passages.
The Catalan Captain was alerted on the morning of the third day of their blockade that the fleet of Alta was approaching their positions. Although the Captain of the Pergamon's fleet was nervous about being outnumbered, he held his fist tightly and ordered his men to prepare their bows and crossbows and organized his ships in a single compact line. The fleet of Admiral Niketas approached the narrow passage with his flagship in the front. Normally, it would've been a foolish decision to attack the better-positioned fleet of Pergamon windward, but the guns and the higher deck of the ships of Alta meant that they would have the advantage, so Admiral Niketas ordered to shorten the sail and angled his ships slightly towards the Chios to the west and ordered his fleet to oar slowly upwind in two lines abreast, a formation in which a number of ships travel side by side, to meet the enemy in the conditions favorable to Pergamon's fleet.
The wind's direction and the slow speed of the approaching fleet of Alta gave the fleet of Pergamon courage and made the individual ships bolder in action, making the lines imbalanced as a few of the ships broke formation and got in a better formation to assault the approaching ships. They weren't wrong to think that the second line of the ships of Alta would be useless, as they had nowhere to flank the enemy. However, they soon realized their mistake once the first line of the fleet of Alta stopped their slow approach with no difficulty and turned their already angled ships sideways, and started bombarding the fleet of Pergamon.
The bewildered Fleet of Pergamon broke cohesion and a gap started to form between the tight formation, as the ships and their captains wanted to distance themselves to avoid the bombardment and the others started to turn and retreat, however, the wind made it impossible for them to escape quickly. The bombardment sunk 4 ships and damaged 10 ships beyond repair, the rest were then approached by the second line of Alta's fleet and were attacked at close proximity with anti-personal cannon fire as well as arrow and a small amount of musket fire before being boarded and taken control of. The flagship of the Captain of the fleet of Pergamon surrendered after his ship was blasted at close range with cannonballs.
With this victory, the siege of Chios began with an overwhelming advantage. The land forces took camp near the harbor and attacked the city walls with the cannons they carried from the ships and the majority of the fleet attacked the castle from the sea with cannons. Since the medieval fortifications were not meant to stand artillery strikes, they were quickly destroyed and before the city was taken in an assault through the gaps, the governor of the city surrendered on the third day of the siege. During the siege, the Almagovars attacked the fleet in the harbor as well as the siege camp, but they were repelled without much difficulty as the harbor was on flat ground and the soldiers were prepared for raids at night. After the city was conquered, the remaining Almogavars surrendered as many of them had their families in the city. Chios went through the same progress as Pergamon did, being incorporated as a territory of Alta, as well as being a naval base.
I looked up a few medieval naval battles as well as early modern encounters of the Portuguese, English, etc. ships that fought against ships with no cannons and in traditional naval warfare. That was the inspiration for this battle and how it played out. I'll add a map of Chios and the passage that I mentioned in the comments.