After the scenes aboard the ship, use the Save Sphere and follow the people into the remains of Kilika. At the bar, head west and you’ll find Yuna, who will be performing the Sending. Whenever that is over, you will regain control the next morning, at the inn. There will be a Save Sphere inside the inn, should you want to use it, but for now, head outside. As you step outside, Datto will run up and say Wakka is looking for you. Head to the right and on the next screen, go to the right of the building there to find Wakka.
Head inside the building in front of you, whenever you regain control, and look behind the NPC on the ground to find Al Bhed Primer Vol. IV. South of that building, on the left side, it seems there’s a chest, but it belongs to the NPC nearby, who actually runs a weapon shop. While you might be tempted to purchase one, it’s best to save your money for now (Wakka will get a new weapon shortly). Double back to the screen with the inn and look for a disheveled building with a little girl inside it on the left side. Stand in front and interact with it to save the young girl, as the building collapses.
Return to the building where you found the Primer to find a chest waiting for you, which contains an Ether. You’re almost done in Kilika, as there’s only one more thing to do. On the previous screen, the one with the inn, there is one other building you can enter, right before the exit. Go inside to find a chest along the left side, which contains three Potions. Now, you can exit Kilika and enter Kilika Woods.