We are faced with the problem that the whole plot of Slaughterhouse-Five goes out of order, so we get the "Complication Stage" of the plot—the firebombing of Dresden—after we see the effects of this firebombing on Billy's mental stability.
So bear with us: we are going to map the progress of the plot as it happens chronologically... but not as it unfolds in the novel itself.
Billy Pilgrim goes to war.
In Chapter 2, we find out that our main character, Billy Pilgrim, is an optometry student in upstate New York who winds up getting drafted to join the army in 1944. He gets sent overseas to Luxembourg to fight the Germans in World War II, but he is taken prisoner nearly immediately.
This POW experience is the thing that sets up both his time-travel and his eventual trips to the planet Tralfamadore.
Billy becomes a prisoner of war.
We learn that Billy becomes a POW in the first sections of Chapter 2, but it isn't until Chapter 3 that we actually see him captured. Billy is carrying no weapons and is on the verge of being shot by fellow American soldier Roland Weary when the soldiers pick him up. Clearly, he is no one's idea of a hero.
Billy is completely under the control of other people throughout the book, and his imprisonment is only the most exaggerated example of his lack of self-determination. Once Billy is taken prisoner, he begins skipping through time: he starts living life out of sequence because he has been so damaged by this initial experience.
Billy witnesses the firebombing of Dresden.
While the specter of Dresden hovers over the entire book, starting with the narrator's introduction in the first chapter, we do not actually see Billy's experience of it until Chapter 8. The sequence of events that changes Billy's life forever—his being drafted and taken prisoner—really reaches its peak here, when Billy survives the firebombing of Dresden.
With no understanding of what is happening around him, Billy suddenly finds himself sheltering underground in a meat locker while an entire city goes up in flames above him.
Billy has a nervous breakdown after the war.
Billy returns to the U.S. three months after the February bombing, after being forced to dig through the ruins of Dresden looking for bodies. We know that Billy's return to upstate New York doesn't exactly mean that he has gone home: he is still skipping through time, and has been ever since he was taken prisoner.
A sense of unresolved mental pain fills Billy's whole story, but it really reaches its climax when he checks himself into a local veteran's hospital to recuperate in Chapter 5, Section 20. In conversation with Eliot Rosewater, another traumatized veteran, Billy learns to escape into the science fiction novels of Kilgore Trout. As the narrator comments, in the wake of all of this emotional suffering, "[Billy and Rosewater] were trying to re-invent themselves and their universe. Science fiction was a big help" (5.20.9).
Billy recovers, but for how long?
We know that Billy goes on to live his life because his story starts in Chapter 2 with the reassurance that after the war he will marry Valencia, have children, and become a successful, well-to-do optometrist. But we also know that he is going to tell the world about his travels to Tralfamadore and have a major falling-out with his daughter.
It is not until Chapter 9 that we finally see the 1967 plane crash that seems to trigger Billy's decision to do something—for the first time in his life—by telling the world about Tralfamadore. So we sit in suspense for many chapters before we find explanations for events that happen at the start of the novel.
Billy begins telling the world that he has been abducted by aliens from the planet Tralfamadore.
In Chapter 9, Billy finally comes to some kind of peace with his memories of the war, being held prisoner, and the Dresden firebombing. But this peace, oddly enough, seems to come once again from outside intervention: a freak plane crash leaves him telling the story of Tralfamadore for the first time.
He informs Bertram Copeland Rumfoord: "Everything is all right, and everybody has to do exactly what he does. I learned that on Tralfamadore" (9.22.10). Billy has found a way to not blame anyone for the pain he has gone through: according to the Tralfamadorians, there is no other way that his life could have happened; there is no such thing as free will. This way of thinking comforts Billy after all of his unresolved pain about the war and his place in it.
Barbara Pilgrim takes Billy's responsibility for his own life away from him.
Ironically, the one thing that gives Billy comfort after he has been quietly suffering for 23 years with the memories of his capture and the massacre is also the thing that leads to his final loss of self-determination and self-control.
When Billy sneaks out of the hospital where he has been recovering from his skull fracture in Chapter 9 and goes on the radio to talk about Tralfamadore, he gives his daughter all the reason she needs to believe that he has gone senile (even though he is only 46). And so now we are back to Chapter 2 and the end of the story (which happens at the beginning): Barbara decides that Billy can no longer care for himself and winds up taking control of his life and business so that he can rest in peace.