Barrie never wrote a single narrative that includes the whole story of Peter Pan. He wrote two stories, which kind of duplicated the same ideas and contradicted each other a little, and he continued revising one of them for years after it debuted. To really get the *whole* story, you'd have to read all the various drafts he wrote of the play.
But you can get pretty much the whole thing by reading two books. The first is "Peter Pan in Kensington Gardens" (1906). This is an excerpt from an earlier book called "The Little White Bird" (1902) (which also included a bunch of material that had nothing to do with Peter Pan). It tells how Peter escaped from the nursery as an infant, met the fairies, etc. The second is Barrie's novel "Peter and Wendy" (1911). This was based on his original script for the play (1904) but includes some additional material, especially the chapter titled "When Wendy Grew Up". It's also been published as "Peter Pan and Wendy" as as just "Peter Pan". It's the story that everyone knows. Between the two of them, this gets you from Peter's first flight as a baby, to the last time Wendy saw him (with some gaps Barrie never filled in).
There have also been sequels and prequels written: Steven Spielberg's "Hook", Dave Barry and Ridley Pearson's Starcatcher books, "Peter Pan in Scarlet", etc. But that's the extent of what Barrie wrote about the character.
P.S. Neither book is under copyright in most of the world. You can read them at the site below.