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[edit] 1st Visit

Upon boarding the Celsius, talk to Shinra. He's the little guy to the right typing at a computer. He can be extremely useful, able to provide dossiers on almost all characters you have met. During the conversation select "Shinra's guide to everything" then "Using Garment grids" to receive the Vanguard Garment Grid and learn how to use it.

Next, at the front of the bridge, go down the path to the right and talk to Brother, on your left, to learn about the Al Bhed language and be taught a few letters. Even if you mastered it in FFX, you still have to collect the Al Bhed primers again. You can also talk to Rikku on your right and Paine, who is on the left side of the area Shinra is in.

Now head up the stairs at the back of the room and save at the Save sphere through the door. Continue along the passage to the lift and choose to go to the Engine room. Open the four chests for items, and return to the lift.

Now go to the cabin and talk to the Hypello at the bar. You can buy items from him, and if you choose to "Rest" at least once in every chapter, you view an extra scene in Chapter 5. Now return to the bridge and walk to the front of the bridge. Confirm your selection and off to Mt Gagazet! (next)

[edit] 2nd Visit

Go straight to Brother and choose to comfort him, then talk to Shinra. He will give you the Black Mage dressphere