There are many worlds, but they share the same sky
Kingdom Hearts starts on Destiny Island, the home of Sora, Kairi, and their best friend, Riku. There isn't much said on how they all arrived on Destiny Island, except that a new girl arrived in Destiny Islands and that she lives in the Mayor's house. A little bit more is explained in the Kingdom Hearts Manga about Kairi.
Kingdom Hearts starts with Sora seeing weird scenes before being woken by Kairi thinking that it was all a dream. Sora enjoys his time hanging with his friends and even starts to build a raft wanting to get off the island and see the other worlds. During these days, Riku teases Sora about the paopu fruit and how if two people share one, they will remain a part of each other's lives forever. It is there that a bit is shown how much Sora likes Kairi for he even competes against Riku to win a chance to share the fruit with her.
One night, strange creatures called Heartless invade the island and kidnap Kairi. Sora searches the island for her and watches as his friend Riku gives into the darkness. He is soon given a weapon called the Keyblade. At first, he is confused but uses to weapon to defeat the creatures around him. That is what starts his journey.
Sora is sent to Tranverse Town and meets Leon, Yuffie, Aerith and Cid. They explain what is going on with the worlds around them. Sora also meets Donald and Goofy who were looking for someone. Donald and Goofy join him and together they travel to every world they come across to find more clues, answer, and information about their friends that they are hoping to find.
Sora travels with his new friends upto Hollow Bastion where he finds Kairi unconscious. He learns that she has no heart and is the seventh princess, the Princess of Heart. To save Kairi, Sora gives up his heart. This wakens Kairi. However, all she can do is watch Sora disappear, but Kairi's feelings were strong for Sora and she breaks the darkness that was surrounding his heart and saves him. Back in Tranvese Town, Kairi gives Sora her lucky charm and makes him promise to return it back to her. This is also a promise that they will be together once again when Sora returns. Sora does return back to Kairi, but they are soon seperated because he still needs to continue his journey and find his best friend, Riku.