As a HUGE note, it is highly recommended you do some of the optional things in this game before bothering with Sin. (See Side quests for the details.) Sin can be difficult to deal with, especially the bosses therein. While you don't need to 100% the Monster Arena or kill all of the Dark Aeons, it is recommended to delve at least a little into obtaining the Celestial Weapons, for example. I mean, if you thought the Spectral Keeper and Yunalesca were difficult, Seymour Omnis and The Final Aeon will do to you what you're about about to do to Sin.
After getting access to the airship and doing as much or as little on the side as you so desire, go speak with Cid on the bridge and opt to go to Sin. There will be a scene, after which you need to reach the top of the ship. Save on the way, because you're fighting something a bit bigger than Evrae this time.
BOSS - Sins Fins
In case you're curious, we're covering both Fins together here due to their huge similarities - it's enough to warrant the same strategy for both, I kid you not.
BOSS ANALYSIS: Both Fins largely rely on the same attacks. They can use a physical attack to all characters - sometimes as a counter - to hit everyone for 1,000~1,500 damage and induce a Delay. There's also Negation to remove the Fin's negative statuses and to remove your positive statuses. Finally, there's Gravija , a move with a one-turn character the rips off 75% of your HP. Pretty simple. They tend to use that basic physical or nothing on the first three turns of their battle routines, then a charge turn, making Gravija occur every five turns.
MAIN STRATEGY : As it so happens, this battle is fought similarly to the Evrae boss - you can tell Cid to move in closer or far off. Being closer puts you at risk for more attacks, but also makes it easier to actually attack. Being far off can help you avoid attacks (i.e. Gravija), but if Cid can't get in a turn... Plus, there's a lot fewer options for distance attacking.
Anyhow, the main strategy is pretty simple. Stay far-off for a few turns to buff yourself up: Hastega, Mighty G(uard) (or Protect/Shell for the main benefits), and then just Cheer/Focus yourself from there until you're ready. If you want to start an offense before pulling in, pull out Wakka or Lulu for their Blitzball or Black Magic, respectively. (Or Yuna's Holy.) When you're ready, pull in close and just wail on the Fin, and I mean blast it with everything you've got. When you see the charge turn for Gravija coming up soon, you can either pull out of the combat for a bit and pull in after, or you can wail on the boss some more, just using Curaga after.
BOSS - Sins Core & Sinspawn Genais
BOSS ANALYSIS: At the beginning, you'll want to mostly focus on the Sinspawn, who is used to absorb magical attacks from hitting Sin. It can use a few attacks, depending in part on whether it is in its shell. Overall, you'll see Thrashing (physical to all), Venom (magical to one; Poisons), Waterga, and Sigh (magical to all, Blinds). It will also Cura itself when in the shell. It actually has a set pattern outside of the shell: Venom, Venom, Thrashing, and a loop from there. Each hits for about 1,250 damage, so look out. Waterga is the counter spell, used when you use magic (other than Demi for some reason).
As for the core, which is Sin? It does nothing. Once the Sinspawn is gone, though... Anyhow, you'll see the basic elemental spells as a counterattack from it; they'll hit everyone. You can only hit the core with Wakka, though, and Lulu after beating Genais. Not much of a point to it anyhow. Meanwhile, it can charge up for Gravija, again ripping 75% of your HP off.
MAIN STRATEGY: Open up again with a nice bit of buffing, mostly Hastega and Mighty Guard (or Protect/Shell on everyone). (In case you thought about it, it's not a good idea to try Reflecting everyone and constantly magic-attacking Genais to try to get a lot of Waterga counters: it absorbs Water.) You'll also possibly want to try to use Silence Buster on it; Silence Buster will hold off the Waterga and Venom, which is nice. Otherwise, a simple offense will work: just wail on it and heal when you need to.
After Genais hits 10,000 HP or lower, it will go into its shell. Every time it is attacked, it will hit itself with Cura, while it will attack with Sigh on regular turns. You'll also begin to see the Core charge for Gravija. Note that Gravija hits Genais, too - attack Genais weakly and it may just come out of its shell (due to the Cura raising it above 10,000 HP) long enough to be cut down to 2,500 HP or a little more by Gravija! Use Curaga from there and then just pound on Genais somehow; probably a small Overdrive or something, I dunno. Whatever you can do.
After this, the Core remains. It will counter with elemental spells at random, but they're pretty minor, rarely exceeding 400~600 damage. If you Reflect everyone, you ought to be fine until Negation hits, if it does. So, with Reflect up, all you'll get hit with is Gravija, which can't kill ... kinda reminiscent of Sinspawn Ammes, eh?
After the battle, speak with everyone and go to the top of the airship again, where you can speak with Yuna. Save on the way to the top of the ship for one last battle.
BOSS - Overdrive Sin
BOSS ANALYSIS : There's not much Overdrive Sin will do to you. The main thing you'll see is some kind of counterattack that can Petrify, Confuse, and Zombify you. The Petrification is the main problem. To be honest, I've only seen it several times out of the several dozen times I've played this game (FFX HD being only like the third time), but it can be problematic without Stoneproof or the willingness to continue. Other than that, you can have 15 free turns, after which Giga-Graviton is used to end the game outright.
MAIN STRATEGY: Open with Wakka, Lulu, and Tidus. Since the former two are likely the only two able to put up a major offense, they can throw Blitzballs and high-level Black Magic. Tidus should be used for Hastega. After that, if Yuna knows Holy, bring her in, but otherwise stick to Cheer (since your offense is likely physical).
After three turns of being drawn to Sin, you can attack directly with your physical units. Switch the main party to Wakka, Auron, and Tidus. Have Auron use Armor Break to halve the beast's Defense, then begin pummelling it with everything you've got. Quick Hit especially is useful here to get yourself additional turns. If you somehow have Doublecast, use Lulu in lieu (lol) of Wakka and begin Doublecasting after a Mental Break. To be honest, there isn't much you can do but attack it for a while and hope for the best. Over the course of 15 turns, you are expected to deal an average of about 9,333 damage per turn, so better pull out what you can.