Autobiography, Postmodern, Science Fiction, War Drama
We could also say "one of the weirdest books we've ever cracked," but that's not a exactly a formal genre. (It is, however, a true statement.)
So let's break down this genre business in a straightforward manner. After all, it's not like Vonnegut is going to be straightforward with us... because the idea of "forward" is a lie, according to his philosophizing toilet-plunger aliens.
Slaughterhouse-Five ain't a pure autobiography because, while it does have elements of the author's life in it, most of the narrative is focused on a fictional character, Billy Pilgrim. At the same time, many of Vonnegut's own experiences in Dresden, Germany, provide the engine for Slaughterhouse-Five's plot... so we think it deserves to be called a semi-autobiographical novel.
Slaughterhouse-Five is also primarily about various aspects of war: (a) how much it sucks, (b) how much it messes people up after it happens, and (c) how generally unfair life is that we have to go fight in wars and then grow old and die afterwards (if we're lucky). So that's why we're also describing Slaughterhouse-Five as a war drama: not only does the plot focus on World War II, but the book also spends a lot of time pondering war as an experience.
As for the science fiction genre, Slaughterhouse-Five uses the elements of science fiction—time-travel and aliens—but it is also self-conscious about considering what science fiction is for. Billy and Eliot Rosewater read science fiction because their own realities no longer make sense to them. They need invented realities that work by different rules because their own lives have totally lost meaning. Slaughterhouse-Five uses science fiction the same way it uses war, both as a plot point and as an object of philosophical examination.
The level of self-consciousness that Slaughterhouse-Five brings to the genres of autobiography, war drama, and science fiction all point to a fourth and final genre: the postmodern novel. The constant confusion about when—or even whether—the different events of the novel happen mean that readers are constantly kept at some distance from Billy Pilgrim and his life story. By using the author as a character in the book and by telling Billy's story out of order, the novel itself keeps reminding us that Billy's story is fiction. This manner of storytelling indicates a degree of skepticism about the idea of a unified self or the possibility of realistic narration that characterizes postmodernism.