For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
Dan, thinking he's happy but secretly fearing death, meets Socrates and, after some back and forth with himself, decides to learn his new teacher's way of life.
Act II
Dan does all sorts of training and learning under Socrates—meditation, rigorous dieting, and the like—for the bulk of the book. But despite all this, he cannot find happiness. He separates from Socrates and lives a mediocre life for several years, wondering where happiness is.
Act III
In the mountains, Dan has a vision of his own death, leading him to understand that nothing can possibly matter and so a person should simply be happy without reason. Soon after, Socrates mysteriously vanishes, Dan marries Joy, and everything is happy… and done.