Walkthrough:Kingdom Hearts/Destiny Islands
When you first begin the game on Destiny Islands, Sora wakes up on the beach, where he and Kairi talk. Sora tells Kairi about the dream he had, after which you are introduced to Riku, Sora's rival and friend. Speak to Kairi (who's by the doorway located on the other end of the area) once gameplay resumes, and she will tell you to get the following items:
All of these can be obtained in this section of the island. One of the logs can be obtained from the shore underneath the bridge. You can get the second log from the island across; use the bridge to get to it. You can find the Rope on the platform on the west end of this side of the island. Lastly, the cloth can be obtained from the tree-house in the middle of this side of the island.
Once you have collected each of the items on Kairi's list (or even while in the process of collecting them), try your luck battling Tidus, Selphie, or Wakka to gain some early experience, or attempt defeating all three at once to receive a Potion.
On the 3-on-1 battle, you should go for Selphie first. Not only because she's the weakest of the three, her special attack will most likely be annoying when around since you can't focus on the others. After you defeat her, move on to Tidus. Tidus may be the most powerful of the trio, but his close-range attack will get in the way when you're trying to chase and beat up Wakka. Wakka's throws will most likely miss you when you're doing combo on someone anyway. After defeating Tidus, defeat Wakka using the same strategy as above.
The other opponent you can face before returning to Kairi is Riku himself, who yields a Potion each time he is defeated (you can repeat the duel as many times as you like to obtain a grand total of 99 Potions).
Head through the doorway where Kairi previously stood to find Riku, who asks you to decide on a name for the raft that he, Sora, and Kairi wish to build. He suggests the name "Highwind," but you are allowed to select anything you wish (the default name of Sora's choosing is "Excalibur"). After you select your name, Riku challenges you to a race, after which Kairi arrives to be the judge. After she explains the rules to you, the race begins.
Again, all of these items can be obtained on the island. The fish can be found swimming in the water in the Seashore. The coconuts can be obtained by hitting the fat palm trees (you're looking for green coconuts, so any brown ones that fall will not be collected). The seagull egg is located at the top of one of the palm trees near the bridge by the door to the Seaside Shack. The mushrooms are scattered around the island. One is behind the rock in an alcove in the Cove. The second is near a post that is also located in the Cove.
The final mushroom is in the Secret Place, but once you obtain it, a cutscene plays showing Sora finishing the Paopu Fruit drawing he started when he was four. When Sora stands, he finds a mysterious cloaked figure standing behind him who speaks of the world being "tied to the darkness" and how it is "soon to be completely eclipsed." The figure insults Sora's intelligence, vanishing as gameplay resumes. Exit the Secret Place and collect the Drinking Water by walking under the waterfall to the left. Now that you have collected all the items, seize the opportunity to duel Tidus, Selphie, and Wakka or race Riku again and locate the hidden Treasure Chest in the Cove, which is located near the wooden tower. Pick up one of the nearby wooden crates and throw it against the wall with a hole in it. Climb up into the hole and open the chest to receive a Protect Chain. Finally, return to Kairi, speak to her to obtain another Hi-Potion.
Once this is complete, return to Kairi and choose to "go home" to begin a cutscene in which Sora listens to Kairi speak of how Riku has changed and how much she can't wait to set sail. Soon after, another cutscene shows the text of the letter Donald previously discovered. Donald and Goofy leave Disney Castle along with Jiminy Cricket after speaking with Minnie and Daisy, determined to find the one possessing the "key" as King Mickey had directed in his letter. Sora is then seen in his room on Destiny Islands, and he suddenly notices a strange storm off in the distance. Sora leaves to investigate, and just when he discovers Riku and Kairi's boats, mysterious black creatures with glowing yellow eyes spawn nearby.
These enemies, called Heartless (the variants spawning in each area are specifically known as Shadows), are immune to physical damage from Sora's wooden sword, so ignore them and run to the bent Paopu tree, where you will find Riku. After another cutscene, Riku vanishes into the darkness and Sora gains a mysterious new weapon known as the Keyblade. You can now attack the Shadows to your heart's content. The Heartless will continue to appear no matter where you go, so this is an excellent opportunity to gain some much-needed experience. Make your way into the Secret Place, where you will find Kairi and the door inside will open, propelling both of you out of the cave and directly into a battle against another Darkside.