When you reach the main HQ for the Operation, go northeast for the scene, after which you can go east to the Save Sphere. You'll also find O'aka there; there's a pretty nasty (for some) boss fight ahead, so prep up. You won't find any ailments or elements (other than Thunder) hitting you in this fight, so don't bother with the Wards or SOSs if you don't have a different reason to. Personally, the shop back at the start of Mushroom Rock is better (often HP +5%).
Anyhow, after that, open the chest to the northeast for a Mega-Potion , then look for the rack of spears more to the southeast. Behind it, semi-visible, you'll find a Serene Bracer . Save at the Save Sphere, then speak with the guard to the southeast to confirm that you're ready. This, of course, leads to a boss!
BOSS - Sinspawn Gui
Okay, so let's discuss the boss as a whole. Your main target in this fight is the boss's lower torso, which has a quaint 12,000 HP. The main problem with this is that the arms of the boss (two total, off to the sides) tend to block physical strikes. The arms are a bit weak at only 800 HP each, but they are also considered Armored, so you need magic or Piercing to ably hurt them. Once they're done with, the main body is open, but only for a few turns.
As for their offense? The arms just sit there. The main body will focus on the usage of a basic physical (500 damage) or Demi, which, like against the Sinspawn Ammes from the start, cuts off 25% of your current HP: it can't kill you, but it will hurt nonetheless. It continually alternates between these two moves, so you can predict what's next. The head of the Gui is the main problem. It can use Thunder, which won't hit too hard (150) due to the head's low Magic. However, when the head starts moving, you need to attack the head, and quick: it is about to use Venom , which can poison and deals around 350~450 damage.
So, what's first? Well, the head is a good start: once that's dead, the boss's largest threat (Venom) is permanently disabled, since the head doesn't regenerate. To hit it, you'll want to use Wakka's Blitzball or Lulu's Magics. You could have Tidus buff up the team with Cheer in this time, or have Auron work on the Arms, or have Yuna sit around to heal damage - whichever you pick for that time is up to you.
After the head's gone (it'll turn grayish), it's time to work on the main body. For this, you'll want to bring out Lulu again: the arms shouldn't be blocking her spells. They will block basic physical strikes, though - bring out Auron and/or Kimahri (whoever has Piercing) so they can chop up the arms. Then, once both arms are gone, bring your strongest out and just start wailing on him. Be sure to hit him with Power Break, though - that will largely aid in reducing the physical damage you take.
The main difficulties in this boss are Demi and Venom. Venom is easily taken care of in a few turns, and Demi with a simple Cura or Mega-Potion from Yuna.
Back from the grave, huh...
BOSS - Sinspawn Gui Part 2
This boss has largely remained unchanged from the previous fight; the main thing is the reduction in HP for the body and head and the near-halved Strength from the main body. Even the offense is about the same. So expect about 200 damage or less from most attacks other than Demi.
You'll be fighting this fight only with Yuna, Auron, and Seymour. Seymour, as you can tell from his simply exorbitant HP and stuff, is pretty much a god (and it's a lot more noticeable later in the game) Trust me, he is a good deal stronger than you, and this is also the only time you'll control him. If you manage, you can let him get hurt enough to use his Overdrive, which will easily hit 3,000+ damage. It's a bit worth it if you've never played this game and thusly never seen Seymour's Overdrive.
Since you're limited to those three, you can quickly guess who to specialize where. Yuna will actually be detrimental to the offense if she Summons (Seymour is better than her Aeons, yes...), so focus on healing with her. Auron is best left to attacking the arms, then the main body, as before. And Seymour just needs to hit the head with a Black Magic spell or two before slaughtering the main body. (Even though this battle would be a LOT quicker if you focus straight on the main body, it's a nice idea, while it is easy to beat up the head and arms for AP and Gil.)
After fighting off the Sinspawn for the second time, there will be a bit of a scene. When you regain control, go north and you'll find another scene to watch. (Who lives and dies in that scene is irrelevant, but it basically is related to whether you told those involved to fight or not before the Operation began.)
After that, go west and save, then speak with everyone. Thereafter, go west some for a Hi-Potion and speak with Auron, then just follow your party as you walk along the road to the Djose region. Shop at O'aka's on the way if you want.