What mini-game? You mean just the concept of sending out ships on missions, or is there more to it? I just started messing with the fleet is why I'm asking
No, just the fleet is the mini game. It has a completely different set of resources and currency (gemstones), so it's a separate mini-game.
If you're looking for pointers, here's what I do: You will build your fleet up initially with schooners, brigs and frigates. Depending on what level you are, man o' wars may be too much right now, unless you waited until late in the story and have upgraded the Jackdaw a good bit.
My final fleet looks something like this, 2 schooners, 2 brigs and then maybe 5 frigates and the rest man o' wars. On a shipping mission, always send the smallest (fastest) ship that has a large enough cargo capacity, so you can save the bigger ones for other missions.
In the battles, there are three slots. The top slot gets a speed boost. Always put a top level (lvl 11, I think) schooner here. It will get off 2 or 3 shots for every one shot from the enemy ships.
The second slot gets a firepower bonus. I usually put a brig here, but a frigate works, too.
The third slot gets an increased resistance to damage, so put your biggest ship here, either a frigate or a man o' war. The enemy usually attacks your bigger ships first, so this one will be a damage sink to protect your smaller, weaker ships.
Once you progress through a few locations, the missions start taking over 8 hours (real-time) to complete, so I usually just do all the battles first and then send out all my ships at once and I'll check it again the next time I'm on.