Besaid Island

Boss: ??? (Kimahri)

You will have seen Kimahri before — in the Chamber of the Fayth and standing around outside the Temple just before Valefor's summoning. He's not particularly dangerous here — his attacks do 50-60 damage, but Jump does about 100-110. Heal if necessary, and use your overdrive, Spiral Cut, if you get the opportunity — it'll do about 350 damage and that's just a little under half his health.

Once you're done, you'll find out who exactly Kimahri is, and he'll walk off again. After some more walking, another preset battle will occur, against a rather large and menacing Garuda. Wakka doesn't want to fight it, so you have to switch Tidus out for Yuna, who then needs to summon Valefor. Once Valefor's out you can begin fighting it — a normal attack does about 200 to the Garuda, but you also have Sonic Wings at your disposal, which will do about half the damage of a normal attack but delay the Garuda's turns somewhat, making it more likely for you to get two turns in between the Garuda. Bear in mind that Sonic Wings won't always work enough to allow an extra turn, and some enemies/bosses are immune to delay. The Garuda has 1400 HP, which will go down pretty quickly. If you want you can wait until you get an Overdrive with Valefor, then use Energy Ray to do upwards of 1000 damage!

After the battle, carry on along the path. You'll run into another Garuda further along — but this time you can use Wakka's Dark Attack to inflict darkness on it. Now almost all of its attacks will miss. Darkness is another standard status effect as far as Final Fantasy games go, and it becomes a lot more common later on in the game. For this battle, attack with Wakka and Tidus (re-use Dark Attack once it wears off) and use any of Lulu's black magic spells to inflict about 250 damage. The Garuda won't last long, although you might want to bring Yuna in and either have her summon Valefor again or just use Cure on someone just so she participates and gets AP at the end. Ignore the random Save Sphere, and carry on along the path back to the beach.