You'll go through a short scene with Tidus looking around the boat to begin with. Once you get control of him, head to the front of the boat to learn a little bit about Yuna. You can talk with Wakka to find out more, as well as an amusing statement from Lulu. She doesn't have much more to say, so head around to the back of the boat and down the stairs. You'll be confronted by O'aka XXIII (23rd), a merchant. It may seem a bit weird and inconsequential to actually lend him any money if you talk to him again, especially after he just commented about your relative lack of money and poor fashion sense, but you need to lend him at least 100 Gil every time you see him. Why? When you get close to the end of the game, and especially from then on as you do the sidequests and such, lending him money now will reduce the prices to which he and his brother sells you items as well as what they actually sell you. Loaning him nothing is a bad idea as he'll charge you more. By the time you loan him more than 10,000 (10,001) Gil, (can be done in little amounts at a time) you'll have a 30% discount.
After speaking with O'aka to the door under the stairs that says "Power Room". You'll be introduced to chocobos. Anyway, check the bottom of the screen here (at the end of the room) to find Al Bhed Primer III. Go back out to the lower deck and enter the door on your left below the Save Sphere. The chest here contains a Remedy, and you can play Kick Someone Else's Property (the suitcase above the door) for a large stock of Potions (if you have less than 20). There are also some of the Besaid Aurochs down here that you can speak to if you want. Head back up to the top deck. There's nothing up the stairs to the balcony, so head to the front of the ship and talk with Yuna.
After speaking with her about Jecht, Tidus' father, the boat will shake. Here's Sin! Or more specifically, Sin's fin. It will close in, and Sinscales will drop onto the deck.
Boss Battle: Sin's Fin and Sinscales
Sin's Fin is the main target here. It has 2,000 HP, and each Sinscale has 200. You need to knock off the 2,000 HP the Fin has to progress the battle, but you won't want to do that straightaway. Each Sinscale you kill will net you AP, so you can theoretically get a huge amount of AP and Sphere Levels from continually killing the Sinscales. Each Sinscale will take two physical attacks from Tidus, Wakka or Kimahri, or one of Lulu's black magic spells. One critical hit from Tidus, Wakka or Kimahri will also do the job nicely. Kill off about 30-50 Sinscales (do this until you acquire overdrive mode Slayer - which fills the overdrive gauge by 20% whenever that character kills an enemy). Although it takes some time to prepare, it is worth the wait. If you use Cheer enough, you can get Tidus, Wakka and Kimahri to one hit all of the Sinscales, making it very easy) then get Lulu and Wakka in the main party and finish off the Fin. If any of the Sinscales' wings start to flicker, make sure you destroy them first since that attack can hurt. Before you finish off the Fin, make sure Wakka and Tidus are healed as well, since you will need to be for the next stage of the fight. Use Yuna's Cure to do this — try to reserve Potions. Don't worry about the Fin either — it's "attacks" will be to move from side to side and replenish the Sinscales once the three on deck are killed. One final thing: do not use Wakka's Element Reels if he gets an Overdrive. Save it for the next battle.
Once you've dealt the damage, Sin's Fin will pull the harpoon gun clean off the ship, and Tidus must have went overboard. Wakka will jump in after him, and help him out. But unfortunately, it's not over yet. A monster appears — time for stage 2!
Boss Battle: Sinspawn Echuilles and Sinscales
Not using Potions in the last battle was important because here it's just Tidus and Wakka with no other methods of healing. If you have Wakka's Overdrive, use it now (element doesn't matter, although Thunder seems to work well) and make sure you get three of the same to hit every enemy. Otherwise, do not bother attacking the Sinscales. They keep re-spawning. Instead, focus on Sinspawn Echuilles. Use Dark Attack to blind it, which will make Drain Touch miss often. Each of the Sinscales will do a sort of torpedo attack which does about 60 damage, and Echuilles' Drain Touch does 60 to one person and heals the damage dealt from himself. Blender, his only other attack, does about 110 damage to both Tidus and Wakka. Between using Dark Attack, just physically attack Sinspawn Echuilles and use Overdrives when you get them, but heal when either of your two characters goes into critical health (yellow). It won't take too long to beat Sinspawn Echuilles. You can do the same with the previous battle and repeatedly kill Sinscales for AP, but this isn't advised since it's a lot easier to die in this battle than the previous one.
After this battle is finished you'll get the rewards — if you did kill a lot of Sinscales, somewhere in the region of 60 Power Spheres, about 1,000 Gil, between 2 and 6 Sphere Levels for everybody, and if you're lucky, some assorted equipment too. The next scene is a sad one, after which you will be off the boat, and in Kilika.