This Airship section is quite long, and could realistically encompass most of the rest of the game; with all the sidequests on offer you're more likely to spend more time on this part and the first part of being inside Sin than you are progressing the storyline.
Thinking One Up
After the scene talk with everyone on the bridge, then make your way out. Note that now all Save Spheres are Level 3, which means any of them can instantly teleport you back to the airship. Talk with Rin to buy some Remedies (this is a generally good investment; Remedies are useful for dealing with pretty much any status effect). Carry on all the way through the Airship until you reach the windowed deck where you first saw Evrae when approaching Bevelle all those game-hours ago. Talk with Yuna and Kimahri. Turns out Kimahri's actually a genius and gives you your first starting point to defeating Sin. Your destination is Bevelle, so head back to the bridge. When you get there, Rikku, Wakka and Lulu will tell you about an idea they had. When you've heard it, talk to Cid and select Highbridge on your map.
Highbridge + Mika
When you arrive, Captain Shelinda (that's right, she's Captain of the Guard) will talk to you and get the guards to stand down. She'll arrange an audience with Maester Mika. When you get there you'll get a little more information on Yu Yevon but it'll mostly be about how you've doomed Spira to destruction by ending the Final Summoning tradition.
The Fayth of Bahamut (the child with the purple hood) will call you to his chamber. Say "I think so." when prompted and pick "Defeat Yu Yevon" next. You'll learn much more from this guy than you did from Mika. Once you've finished talking with him, you'll have a pretty good idea of how to go about fighting Sin. Cpt. Shelinda will reappear on the Highbridge with your party apologizing for the fact that she can't find Mika (and you know why). Rikku will then tell her to get everyone in Spira to sing the Hymn when they hear it. Having a high-up official in Yevon listing to your wishes really helps, doesn't it?
You'll be on the airship again once you're done. Talk with everyone on the bridge and then you can talk to Cid. A new location labelled 'Sin' will be on the location select screen; be careful about picking this option as it begins a long string of fairly tricky battles before you can punch a hole in Sin and get inside. It's up to you whether you begin some of the easier, shorter side-quests first to strengthen your party a bit, or go straight in to fighting Sin. You're strong enough either way, but the sidequests make it easier.
Regardless of whether you choose to do sidequests first, this guide will now continue from when you select 'Sin' as the location and start the fight. Finish talking to Cid and head on through the ship. Save when you get to the viewing deck and then head up the lift and out onto the outside of the airship. You'll get a glimpse of Sin's ultimate attack; which wreaks massive devastation upon the entire planet. After that, Cid will spot a weak spot at the base of Sin's arms — so you'll move in.
Boss Battle: Sin
Stage 1: The Left Fin
This isn't a particularly difficult battle. The weak spot itself has 65,000 HP which is not that much, but first you've got to get up close and personal to inflict damage. Give the command to Move In with either Tidus or Rikku, then get out Lulu, Wakka and Yuna to cast spells and throw balls at Sin until Cid makes his move. Use Haste as well, as the weak spot is immune to Slow and Delay. Once up close, use Armor Break so that all your normal attacks do proper damage again. Keep using attacks on it until the weak spot begins to charge; its now readying Gravija, which will knock off 3/4 of a character's current HP (like a much more powerful version of Demi). Use the trigger command Pull Back with either Tidus or Rikku to pull back and avoid it before it hits (like pulling away from Evrae before it used Poison Breath, except Sin won't try nearly as hard to make your life hell). Other than this, Sin has a ramming attack which does about 1000-1500 damage to everyone, and Negation which it may use as a counter-attack which takes all positive status effects off you and all negative ones off Sin. At distance, Sin will mostly "remain motionless" although will swipe his fin across the deck, dealing about 1500 damage to everyone.
When you've dealt enough damage, you'll pull back and Cid will unveil the Airship's secret weapon — two giant energy lances. These'll blow Sin's left fin off completely; now, you'll have to tear down the other side.
Stage 2: The Right Fin
Nothing's different about this battle; the weak spot has the same attacks and the same HP, so use the same tactics as before; attack with Holy, other magic and Wakka's ball until you're close up and then use Armor Break, then keep attacking until it charges up; pull away, and repeat until you've won. Cid will blow Sin's other fin off, then tell you that the cannon's busted and you've gotta try something else. But, Tidus is one step ahead of the game and decides that you're going in. You'll do a remarkably accurate free fall at high-speed onto Sin's skin, and land by another weak spot. Unfortunately, Sin calls in some help this time.
Stage 3: Sinspawn Genais and Sin
You're up against two bosses at the same time now; the first is Sinspawn Genais, which looks exactly like Sinspawn Genaux from Kilika Woods; except it's more powerful and has 20,000 HP. The core behind it has 36,000 HP. You can't directly hit the core until you've dealt with Genais; however it might decide to retreat into its shell, and will counter-attack anything with Cura which replenishes about 1500 health. This also allows the core to charge up Gravija. However, this time, Gravija is inescapable and is a lot more deadly since Genais can then follow it up with an attack. Genais might also use Sigh, too; this does about 1000 damage to everyone and inflicts Darkness. DO NOT use Brotherhood on Genais, it will heal it.
Genais' Venom will do about 1600 damage and inflict Poison, and that's pretty much the only attack it'll use; it's got slightly more powerful versions of Genaux's attacks as well. Wear down Genais and attack Sin with magic whenever it goes inside its shell. The core will mostly remain inactive for most of its turns, only doing something when Genais retreats inside its shell. When it does, just harass the core instead with Holy; it may counter-attack with a weak black magic elemental spell that hits everyone for about 500 damage.
Once Genais is down, you'll get nearer the core and now you can start using physical attacks. Begin with Armor Break and just pound it until it dies. You'll get a brief opportunity to spend all the sphere levels you've obtained on the bridge, so do so and then save. The next stage of Sin won't begin for a while, so talk to all the party and make your way back through the airship. Go back up to the viewing deck and up onto the outside of the airship, where you'll meet Yuna outside. After the talk, Sin will grow some white wings and fly back up into the sky. Now it's crunch time. Head back through the airship once again and head out onto the outside deck of the airship.
Stage 4: Sin Overdrive
Sin has an overdrive. Sin's overdrive hurts. If it gets to use it, which it will after its 13th turn, it's not just instant death to everyone; it's automatic Game Over. Make sure to have everyone in your party have armor with the Auto-Med ability to save you from certain status ailments.
You do not have time to mess around in this battle; you need to do a lot of damage (a whopping 140,000 HP) in a short amount of time. Haste is absolutely necessary. Have Tidus cast it on everybody if you get the chance, otherwise just on Yuna and Auron will be enough. After you are "Drawn to Sin" for the 3rd time you will be within physical attack range; at this point, cast as many Aeon overdrives (or use Holy) as possible with Yuna then switch her out for Wakka or Tidus. Once you're as close as you can get to Sin, some of its turns it might use a energy blast, called Demi, which inflicts about 600 damage and occasionally Confusion, curse or Zombie. Have Auron use Armor and Mental Break and just keep pounding Sin. Eventually you'll beat it and you'll be able to head inside; it's like a whole different world. But a familiar voice fills the air before you land.