Eventually, you’ll end up on the deck of some Salvage Ship. There, speak with the girl from earlier and she’ll speak about the Sphere Grid . As you probably already have gathered, there are two Sphere Grids in FFX HD (and, really, the PAL/International versions of the original PS2 release). The original Sphere Grid models the original, "easy" Sphere Grid American FFX players had before. It divided the characters into their specialized paths (like Tidus for fast, physical attacking) before forcing them to go elsewhere. The Expert Sphere Grid from which you could have also chosen looks different mostly because of the sheer freedom you can have: every character could become a Tidus if you dared, for example.
In any case… You’ll go through the tutorial on how to use it. Basically, earning AP earns you Sphere Levels (SLVLs), which is currency for moving over the Sphere Grid (1 SLVL for untraveled paths, one node at a time, or four nodes on traveled paths). You’ll get to use Spheres to place nodes or activate nodes to earn their benefits, such as stat boosts or new abilities. That’s basically it. Be sure to use it yourself!
After, speak with the northwestern guy on the ship for three Potions , then examine the whitish object on the deck nearby to find Al Bhed Primer Vol. I - this will fill in some of the letters in Al Bhed conservation. You usually have to grab about 15 or so to be able to truly fill in the blanks, though. Anyhow, speak with the girl who gave the tutorial again to dive underwater.
As a notable footnote, when you dive in the water, you can approach the Salvage Ship again and press the Square Button to be let back on it. There’s a slight bit of misprogramming (I think) with this sequence, in that you can speak with the guy who gave you the three Potions again for three more Potions, then do it again ad infinitum until you’re full of Potions. It shouldn’t take much more than half an hour to grab 99 (33 trips), though their utility is pretty bad - 50 or so (17 trips) will suffice for the while until you get more effective means of healing but really you can get by with only a handful of Potions.
As a footnote to this footnote, though, I’ve gotten word that this glitch does not work in versions released in the U.K.