Montag's desire to read books is a slow process that gradually builds to the actual act of reading one. At the beginning of the story, he already has a book hid in his house, because as he walks into the house the first time we meet him,
"he stood looking up at the ventilator grille in the hall and suddenly remembered that something lay hidden behind the grille."
He only looks up there after he pondered Clarisse's question of, "Are you happy?" So here, Bradbury seems to be alluding to the fact that Montag was not happy, and his unhappiness has led him to books; perhaps he feels that they hold the answer. His desire to read books is there even before the novel begins. However, Clarisse sparks his interest more, along with Mildred's suicided attempt. He realizes, through these two events, that he indeed is not happy, and neither is his wife, or most people that he knows, for that matter. But Clarisse is; she's happy, and perhaps books are the reason why.
While all of this is mulling around in his head, he is called to burn Mrs. Blake's house. Mrs. Blake, rather than leaving her books, chooses to be burnt alive with them. This really gets to Montag; he thinks that if someone is willing to die for books, then there HAS to be something in them, there just has to be. So, this adds to his interest. Then is the visit from Beatty, where he explains the entire history of firemen, and how books became unpopular. Then at the end of his lecture, he basically gives Montag permission to read: "We let the fireman keep the book twenty-four hours." Montag takes full advantage of that time-frame, and tries to read with Mildred. He lists all of the things wrong in the world, and tells Millie, "An hour, a day, two hours, with these books, and maybe..." thinking that somehow they hold the answer to life's miseries. It is a pretty futile process though, as Mildred complains the entire time and is upset that he isn't going to work. Frustrated with her and her friends, he seeks out Faber, who confirms everything he has been suspecting about books. From there on out, he has made his decision; books are worth fighting for, and he takes that conviction to the end of the book.
So, through a slow, gradual process, Montag goes from secretly hiding a book but never reading it, to wanting to read it, to openly reading it, to planning subterfuge in order to bring books back, then open rebellion in the name of books.