I know there is a guide out there that is supposed to say which ship are the best for Kenway's fleet minigames, but since I kindda played around trying to capture all types of ship myself, I'll let you know which one are the best per types, how to find them and what I believe are the best ship to have in your 15 slots !
First the schooners, the ones you should capture are the lvl 11 schooners, which u can find around convoys or in the middle part of the map. They are the fastest of all the ship u can capture, have a 20 units cargo hold and can be used against any type of "hazardous" fights on your routes. They usually shoot faster than most gunboats (and oneshot them) and can shoot twice a lone schooner before it shoots back. I'd suggest having two or three once you've unlocked all the routes and use them for the most lucrative missions with low cargo needs. Try and keep one available for all the low level fights.
Then comes the brigs. Those are probably the best fighting ships around to take on frigates / brigs compositions. There are three levels of those around the world (17, 20 and 25) and though I'm not sure whether the level 20 and 25 are equals in stats once u capture them, just go for the level 25 (found in the middle as well) and capture those. If u are far in the game and ur Jackdaw is pretty strong, u can get lvl 3 heat and the bounty hunter patrols that will come after u will be made of one such lvl 25 brig and lvl 38 frigate.
It can be used for a few missions requiring 35 cargo space (that's it max) and it will go faster than any frigates. However, I suggest u capture 3 or 4 and use them mostly to take out the dangerous fights on your routes.
Third are the frigates. Same here, 3 levels roam the world, the lvl 23, 29 and 38. Weirdly enough, the lvl 29 and 38 aren't different and so there is not much point capturing the lvl 38 frigates (which are deadly midgame to a poorly fitted Jackdaw). I read on this forum that the written guide mentionned they were supposed to be better, but they aren't. Is it a glitch or did they change ? No clue. Those frigates will have 60 cargo space and they'll constitute the bulk of your travelling fleet as they go faster than the men'o'war and the cargo they can carry is enough for most of the late routes. You should have 5 or 6 in your roster and maybe keep one available for the bigger red fights.
Finally, the men'o'war are separated in three categories (not countain the legendaries which are not boardable) : lv 36, 49 and 60. This time, there is a rather large difference in speed and cargo between the lv 49 and 60 and the lv 60 are the ones you should aim for. They are fairly rare however. U can find some guarding the islands south of the world (they do not roam around so be sure to capture those) or by reaching the max bounty hunter lvl, in which case a lv 60 man'o'war will come chasing u together with a lv 38 frigates. I suggest u fill the rest of ur slots with those (4 or 5) to use for ur > 60 cargo missions and keep one available for the hardest fight on the European routes.
So in the end, here's what you should capture if you go for a spree to renew / start your fleet : Two lv11 schooners, Four lv25 brigs, Five lv29 / lv38 frigates, Four lv60 Men'o'War. And with that, you should be all set to do most missions and handle any fight comfortably.
Hope this helped you lot knowing what you should aim for !