We have a lot of stories but very little solid fact about the Chinese naval warfare in this period. My answer will be taken from Rafe de Crespigny’s books: Impreial Warlord and General of the South. He made the same argument in both 2 books abour Three Kingdoms era’s naval warfare:

1. What is the purpose of naval warefare?
In Three Kingdom period, “naval warfare” usually means river warefare exclusively, as there is little fighting in the sea. There might be pirate hunting operations to secure the coast, but large scales conflict were reserved to river fleets wrestling the control of the inland wateway. The defining characteristic of river warefare is its limited area. Usually, you would know exactly where your enemy is and your land army is often close enough to support each other. Because of that and Chinese warships are not designed to sink enemy vessels (more on that later), the navy is often an extension of the land army with primary goals is to support land based operation rather than being an independent forces.
A naval fleet secruing the river, first and foremost, is to facilitate the easy transportation of troops and supplies. On land, the supply line depend on beasts of burden and human’s labor. The beasts and laborers all have to eat, they are vulnerable to disease and enemy’s raid. Therefore, large scale transport by big ship is always prefereable if it can be done. When Yuan Shao marched south to Guandu, his supply warehouse was setup in WuChao and Gushi, both closed to a river; when these depots were destroyed, his army fell apart. When Cao Cao marched north to White Wolf mountain, eventhough he was fighting a weaker force; he must first ordered the construction of 2 canals just so his ships can bring much needed supplies to the far frontier. On the other hand, Zhuge Liang’s land locked northen expeditions suffered from repeating shortage of food and always ended prematurely. Only until they started using some sort of inovated cart (Wooden ox) was the Shu army is able to wage war for more than a few months.
Controling the river also allows much better mobility of troops. Firstly, it allows your troops to cross the river at will. A river is a natural defensive moat, naval dominance will ensure your troops can cross it without being slaughtered. Secondly, it allows your troop to attack and retreat with freedom. For example, when Wu admiral Lu Meng proposed the construction of some fortification along the border; his officers argued against it, quoting the usual protocol: taking advantage of their uncontested control in the YangTze river to disembark anywhere they need, fight, then retreat to their ships. The ships will be their mobile fortress, where they can strike from and withdraw to – therefore traditional land based fortifications were unecessary.

2. But then what kind of vessel did they use? And how did ship to ship combat happen?
What we know is not very concrete. The book Taibo yinjing about warship of the Tang dynasty categorized them (Tang warship) into 6 different types, 3 of which had their name poped up different text of in Han & Three kingdom era. Those 3 were the louchuan (Towered ship), the doujian (Fighting ship) and the mengchong (Armored charger). And actually, we don’t really know what they exactly were. Taibo yinjing seem to be contradicting the account from the Record of three kingdoms at various point when it comes to these ship. Therefore we could be dealing with a literal case of Theseus ship which the ship name remained but their designs and purposes had changed during the centuries.
The mengchong is a curious vessel. It name implies it get extra protection, probably by a leather cover, and it purpose is to charge. The Tang source describe this as a light, narrow and fast ship with the purpose of charging and breaking the enemy formation. The text Shi ming by Liu Xi in 2nd century AD aslo described it as a ramming vessel. Record of three kindom, however described it as a floating fortress which can house hundred of archers. During the Sun Quan – Liu Biao conflict in 208, Liu Biao’s admiral Hoang Zu anchored 2 mengchong to block a choke point in Han river, a thousand crossbowmen on top of these 2 ships rained fire on the approaching Sun army. But Sun’s officer Ling Tong & Dong Xi lead a hundred troops wearing two layers of armor, sailed straight at the mengchong, cut their anchor. And thus the great warships lost control and were swiped downsteam, probably overan and captured by the Sun army.
The doujian, according to our Tang source, is a large open ship with bulwark and loop hole for archer and spear. Oddly enough, it is said to be designed for “combat” while the mengchong isn’t. This could be interpreted as the doujian is designed for boarding action and melee combat while the mengchong is used in ship-to-ship engagement. We can make a guess that in Three Kingdom era, the huge mengchong will lead the way, use its size and enormous momentum to breakup enemy formation so smaller doujian can board the isolated enemies.
The louchuan, as far as our Tang source is concerned, it is a big vessel with multiple decks as firing turret, hence the word “tower”. A “Louchuan office” had been around since Former Han dynasty, in charge of building warships and training marine fighters – so it was probably the signature vessel of the Han era. However Record of three kingdoms rarely mention this vessel, using the words doujian and mengchong instead. De Crespigny argued that although Tang source considered the louchuan a distinct class, it may be not be the case in Three kingdom period; and louchuan is only a term for general purpose warship with multiple decks.
While the mengchong is a ramming ship, and there are other mentions of Chinese warship ramming in battles, it is probably not the main tactical goal of a Chinese river battle. Mediterranean warships were designed to ram and punch a hole underwater – which would sink the enemy vessel. Chinese ship building on other hand often assemble the ship from different compartments to keep it afloat even if one section was penetrated. Underwater ram might have been used but rarely, big projectile launcher like caltapult is also not present on ship as far as we know. That’s is another way of saying that 3 kingdom warship is actually not very good at destroying other ships. The dirty business will be by boarding and arrows where the ship would act as floating platform for melee soldiers and archers. Or setting your own ships on fire and sailing them to the other bank, that can works too.

Source:
De Crespigny, 2010, Imperial Warlord
De Crespigny, 1990, General of the South