radiant gardenWhilst it is revealed early in Kingdom Hearts that Kairi's home world is not Destiny Islands, it isn't resolved until late into the game exactly what world she is from. Sora discovers that Kairi's true residence was Hollow Bastion (even then, it's not until sometime into Kingdom Hearts II that Hollow Bastion is restored and the name – Radiant Garden – is remembered by its inhabitants). Though Kairi does get to go there, she doesn't stick around, though this is not exactly by choice, since she and Sora have to leave in a hurry back to Traverse Town in order to recoup, which is when she stays whilst Sora goes to battle "Ansem". Kairi's amnesia prevents her from remembering her original world for almost ten years, until Sora recalls the memory. The memory Sora sees is of a young version of Kairi, about four, being told her favourite story from her grandma about the light and how it survives in the hearts of children. At first, Sora doesn't know where exactly Kairi was in this memory – until he personally finds himself inside the library he saw Kairi in.
During the events of Birth By Sleep, Kairi was still living in Radiant Garden, which is where she encounters Aqua after some Unversed target her. After saving her, her grandmother comes to find her. Kairi's other family and parents are not mentioned - it is likely she doesn't have any, and the current status of her grandmother is unknown. It's possible she did not survive the destruction of the world, however, and Kairi doesn't appear to know what happened to her or had the opportunity to find out, as of yet. Whilst the other residents departed Radiant Garden via gummi ship which eventually led them to Traverse Town, the charm cast upon Kairi by Aqua beckoned her away to Destiny Islands. Since, the only time she returned to her home world was when Riku brought her back there to unlock the final keyhole.
It's unclear if Kairi knew the other residents whilst in Radiant Garden as a child, though likely; whether they would remember her is another matter, but she was present in the world at the same time as Aerith, Leon, Yuffie, Cid, and Scrooge McDuck. As well as these characters, she would have perhaps known Lea and Isa, as well as Ansem the Wise's other apprentices. It's plausible that Xehanort may have been interested in her, as she had no darkness within her heart. In the later games, Kairi does interact with several members of the Organization, so, the Nobodies of these individuals, but shows no recognition towards them, probably due to her age. But it's not something that can be ruled out entirely.
Personally, I think we'll be seeing a lot more of Radiant Garden and Kairi in the future, as if anyone should go there to help the ailing Nobodies that are now human once again, it's Kairi and Lea. She's way overdue a homecoming.