Rather than employing a conventional third-person ‘‘narrative voice,’’ the novel is narrated by the author himself. (eNotes)
The novel is narrated by "Kurt Vonnegut". We will put his name in quotes here because the narrating voice is identified with Vonnegut as the author but also is associated with Vonnegut as a character in the story.
The narrator speaks directly to the reader through the first chapter then tells the story of Billy Pilgrim. During Billy's story, the narrator inserts references to himself as a figure in the war on at least two occasions.
There is only one speaker then, but this speaker switches from first person narration to third person narration when he begins to tell Billy's story. This method of narration is unorthodox and serves to complicate the text in ways that support the themes of 1) the importance of perspective and 2) story-telling, not as re-telling, but as a creative act.