I'm answering this question taking into account how you could use the environment with a stone age equivalent knowledge to help you in your attack/defence with a little bit of fishing thrown in. these are the sorts of things that you could learn from trial and error and don't need any futuristic tech or knowledge base to decipher.
As on land, Use nature to guide you in your underwater tactics. It's pretty much already solved most of the tactics aside from weaponry.
Breathing and air supply are not an issue. But hey, even whales need to breach every now and then, so if necessary take that into account.
You have some sort of thermal insulation from the cold.
I’ve listed several things to take into account
1) Light & Darkness = One very important factor to take into account. Little to no light reaches beyond about 100-200m of the ocean surface So you could either use the darkness to make surprise attacks or retreats from below or you could force your opponent into the darkness to disorient. For this you would need some sort of light source or visual enhancer for yourself.
2) Bioluminescence = Nature has several bioluminescent species which create their own light. You could bottle up a couple bright sources. Either use them as 'torches' or if they are particularly bright (it's a make-believe world, so you can make them as bright as you want) you could cover them up, take them into a battle in the dark depths and then suddenly unleash a light bomb on your unsuspecting enemies. Making either an attack or retreat under the cover of ...lightness!
3) Water pressure = The deeper you go, the more the pressure. So long wavy spears won't work in the deeper depths and close combat swimming would be the order of the day. Whereas it could be the opposite in the shallow depths where it is easier to wave a long stick through the water.
4) Speed = I'm not sure how water pressure would affect projectiles. Maybe make shooting them upwards easier and could be considered the equivalent of the high ground. I know from experience that swimming underwater is easier a few metres beneath the surface where the pressure is more uniform around you. Swimming would be faster deeper down, but large quick movements slower. Stabbing would be easier than slashing, think more Zulu assegai than dancing twirling spears and halberds (axes on sticks).
5) The Bends = Water pressure could be used to your advantage by forcing your enemies to rise too quickly and get the bends; Nasty way to go. Or at least too much pressure on the ears. Think in terms of mammoth hunters tagging each other out when they got tired of running (ie you need to stop and decompress yourself). This would involve preplanning and groups of people waiting at designated depths.
6) Extreme temp variations = Some ocean valleys have hydrothermal vents, with super hot water; Force your enemies to either burn or flee. The same in the polar regions. There is a rare event where seawater freezes and the remaining denser saltier water forms a downward 'tendril' or brinicle reaching for the seabed freezing everything it makes contact with. It was called an Icy Finger of Death. I must say it is one fantastically magical looking naturally occurring event.While we would be fast enough to avoid getting stuck in it (it takes about 12 hours in real time to form), it could catch a few arms, legs, tails unawares, if you were distracted or trap a wounded unconscious soul. If you are just hunting for food, it could be a cool drive by - pun intended. Just chip out want you want and move on.
7) Currents = Use the currents to your advantage. If you fighting in the depths you could force your enemies into an upwelling current, and again they die of the bends if they can't get out quick enough. Or you can use a downwelling current to quickly escape/retreat. Get yourself away from your enemy, and then take your time to rise back up to the surface when you feel safe! You could string up some huge nets (material of your choice) downstream and then fight and drift downstream guiding your enemies into the nets.
8) Sight= Warmer, slower currents are normally sediment laden, so visibility will be poor. Colder upwelling water is filled with nutrients and is therefore teeming with life. That's why the fisheries are always located in those regions. So you can use the wildlife to your advantage to distract/ hide in/ lure in some predators or just easy fishing etc.
9) Sealife patterns = Ocean wildlife also has an oscillating pattern according to the day night system. Depending on the species they will rise to the surface and sink to the depths during the night and day to take advantage of light and nutrients. Some of this is passive like, phytoplankton. Teeny tiny bits of grub feeding off other teeny tiny bits of grub. Others use their own propulsion to take advantage of the rich easy food pickings. Think jellies. Masses of stinging jellies. You could time your attacks to coincide or avoid, whichever floats your boat.
10) Also think BLOOD = Are there any sharks around. They are apparently the bloodhounds of the sea - I wouldn't want to stick around a battle/hunting location too long. You would need some sort of substance to cover any wounds and wrap kills in that will prevent the sharks following you home. Unless you like having lots of sharks right outside your front door or that is your intention to lure them somewhere. I'm thinking some sort of waterproof gel-like slime or fish mucus etc - something that sticks and seals at the same time and/or some sort of sealed container for any bloody kills.
All of these features would and could be used to their advantage by a people living in the stone age. It's just up to trial and error for your swimmers to figure it out. The world is your oyster!
I would highly recommend watching the BBC's Blue Planet and Frozen Planet for how ideas on how the natural world works and use that to inspire tactics that would work in those situations.