NOTE: While this is the time to do some Side Quests, keep in mind that if you obtain more summons, you'll make the last battle tougher.
When you head outside you'll see Sin...and you'll later be given control of the airship(finally). Head to the rear of the airship and talk with Yuna. You'll soon learn that you have to find Mika, the very one that betrayed you earlier and make him talk. But as you head back to the bridge the plan changes completely. Go to the highbridge in Bevelle and talk with Mika. Then you'll talk with the Fayth.
When you talk with Cid, enter the passwords: GODHAND, Victorious, and MURASAME. Make sure to input them in capital letters. The passwords will reveal new locations scattered within Spira. These are obviously optional, so they will be explained more in the Side Areas section. Select the Sin destination and continue on. Afterwards go to the back of the ship and go through the same path you went earlier to fight Evrae.
---Left and Right Fin---
This battle really isn't too hard, depending on how you've leveled up. You will first start off at a distance...so using Aeons' special attacks, Wakka's Blitzball, and Lulu's black magic is the key. Take this time to build up your Aeons' Overdrives by keeping your distance and using their special attacks. When attacking from a distance cast Protect on everyone. When up close use Mental Break and Armor Break against it. Watch for it to charge it's core, that means a magic based attack, so anyone with Shell will take half damage. After Gravija hits, it'll take 3/4 of your HP away. Be sure to use Mega Potion or Curaga to heal.
---Sin, Sinspawn Genais---
First take out Sinspawn Genais...you won't be able to damage Sin unless you take out Sinspawn Genais first. The Sigh attack wil do about 1200 HP of damage, but use Protect to cut it down to 50%. When Genais covers up it'll counter attack physical attacks with Cura, so don't bother trying to damage it...unless you use Auron. Cast Hastega and Shell to have the Gravija attack take down 3/8 of your HP instead of 3/4. The boss will cast low level magic, sometimes after Gravija. This will mean instant death for characters if your HP isn't at least 680. Try to avoid using Overdrives, this boss isn't too tough for you to use them. When the core charges and your HP is low, don't bother to heal until after Gravija. Since Gravija is a gravity attack, it cannot kill you.
After the battle save your game. While those two battles were long, they were relatively easy...but don't expect that for long once you're inside. Regular enemies can pose tougher challenges than the two bosses you just faced.
When you're on the airship, head to the back of the airship. Talk with Yuna, and you'll later head back to the bridge. Afterwards just head to where you last spoke with Yuna.
Sin's Overdrive is an obvious killer. First cast Haste, Shell, and Protect on all party members. When it gets close, use Mental and Armor break. It will use a Demi attack every other round, so don't worry about healing. When it's Overdrive is about to hit, make sure to bring in a Aeon and use Shield. Don't bother to cure your status ailments unless you need to heal since it will just keep casting the demi-status-effects attack. Your party should be able to knock the boss down in time before it casts it's Overdrive though. Even though it has 140000 HP, you should be strong enough to deal significant damage to it. It's not that hard of a boss battle believe it or not.
I suggest you avoid fighting King Behemoths, they'll cast Meteor when they die and most likely it'll kill your party members. Now equip your party with armor that absorbs or nullifies elemental attacks...any one of them, preferably an armor that Wards off 3 to 4 elements.
Seymour will use Firaga, Blizzaga, Waterga, and Thundaga pretty much every round and up to four times in a row in this battle. For a summoning tip, make sure your Summons are all in Overdrive mode. Avoid summoning an Aeon that is using the same element Seymour is using...the summon will most likely be banished...but sometimes not. Only use Aeons for their Overdrives in this battle. Sometimes if you have an Aeon matching the same element Seymour is using they'll be healed every round...but you must attack regularly or use your Aeons' special attacks. Seymour will cast Ultima everytime after Dispel...so if you can, cast Shell on a remaining character or have an Aeon Shield it. Use all neccessary Overdrives in this battle and keep your HP up. Have Kimahri use Mighty Guard in this battle...it comes extremely handy since it casts Shell, Protect, and NulAll.
After the battle go back and save your game and continue on. Now the enemies get a bit tougher here. You need to kill a certain number of fiends to open each door...now that's asking a lot. If you ask me, and if you've been leveling up, you don't need to open the doors just yet. You can continue on ahead to the next save point.
Now when you see a giant structure fall, touch the emblem. Now in this next area are several crystals that shoot up. Make sure not to touch them, otherwise you'll be put in a battle...but you can run from the battles. There are 10 items that you must pick up, all of them very useful. Pick up those stationary crystals and after 10, the last battle shall begin. Make sure to equip armor that protects against stone status too.
---Braska's Final Aeon---
Now the key to winning this battle is to talk when the Overdrive is about to build. Try to save your summons for the next form. If you can inflict Zombie status on Jecht, then do so. Otherwise hit the Pagodas once in a while to stop their movement. You won't be able to destroy them since they keep coming back. If you can, use your Aeons to use Shield(but not their Overdrives) just before Jecht's Overdrive. Although when you talk with Jecht and his Overdrive doesn't stop building, don't worry about it. As long as you talk to him the Overdrive will start over. But be careful though, you won't be able to use the Talk command every time. Just save it until the Overdrive starts to build. Use Armor and Mental break and cast Protect and Haste on as many characters as possible. If the Overdrives initiates, Jecht Bomber will hit, taking away 3/4's of the party's HP.
Note that each time you take down the Pagodas you'll have to damage them even more to take them down the next time. Using a Zombie status will help tremendously in this battle. Watch for Jecht's Beam...it will petrify a character and then put that character out of battle...leaving you with one less party member. Now in the second form cast Protect, Shell, Haste, and Regen continously and heal any negative status effects. Hopefully you've saved your Overdrives till this battle. I don't think I have to say much for the second form, and if you have to do anything desperate...then do it...because this is the last battle. Use Zombie Attack again on Jecht...watch for the Pagodas, if you can't use a Zombie status on Jecht, good luck winning this battle. Note that you must use the Zombie Attack continously...it wears off everytime the Pagoda tries to heal him.
Now the next battle, well...it really isn't a battle, you will have to call your Aeons and defeat them one by one. They only have as much HP as you initially began them with prior to the battle with Jecht. You'll be surprised by how much damage you do since they have high defense and magic defense, much like Yuna's stats tenfold. You really can't die either, you'll be revived. You can summon other Aeons since Yu Yevon cannot possess two Aeons at the same time. Use a Zombie attack on Yu Yevon(99999), otherwise you won't stand a chance. Use it after the Pagodas heal him, and when it tries to use Osmose in Zombie mode, it'll actually give you MP since undead characters hurt themselves draining living people. Once again you won't die...Auto-Life is always on. Since this battle is so easy, I don't really call it a battle since you can't die from it.