An initiation story typically concerns a young person who is "initiated" into some adult reality previously unknown to him or her. Stories like "Through the Tunnel" and novels like A Separate Peace and Lord of the Flies come readily to mind. "The Cask of Amontillado," however, could be considered an initiation story although it does not seem typical of the genre.
Fortunato is, in fact, initiated into a reality he did not know or understand, until it is too late. Through his relationship with Montresor, he discovers the reality of evil that can live in others. Fortunato learns the reality of Montresor's rage, deceit, deviousness, manipulation, and madness in all its fury as he realizes he has been lured to his death by one he considered a friend. As Fortunato realizes he is being walled up--buried alive--by Montresor, his initiation is complete, but he will not live to benefit from his deeper understanding of human nature.