What is the moral of this story?
To me, the moral of this story is that people need to fight to keep their humanity. If they do not care about intellectual and emotional life, it will be taken away from them.
In this story, Guy Montag and people like him have very unhappy lives. They do not have serious thoughts and they do not have emotional relationships. Why is this? It is because people have stopped caring about stuff like that. People have chosen mindless entertainment over things that will make them think. They have chosen to watch the parlor walls and things like that instead of hanging out with friends and family in the real world.
So Bradbury is warning us -- if you don't think, you will lose the option of thinking. If you just watch TV and stuff instead of interacting with people, no one will care about each other. He is telling us that we need to think and to care about other people or we will end up in a dystopian world like Montag's.
I think another theme is alienation and loneliness. Even though Montag lives with his wife in a large city and has an important job, he really is alone. His wife spends her time plugged into the TV walls living vicariously through them. Montag doesn't really have his loneliness is broken until he meets Clarisse. She makes him realize how alienated he has become to the world around him. He takes being alive for granted. Faber too helps relieve some of this. Of course, when Montag begins to smuggle books into his home, he becomes further alienated from society.
However, after Montag has fled the city and it has been destroyed, he has his alienation relieved by his initiation into the book people. While he has lost so very much by the novel's resolution, Montag has at last found a real purpose to his life, that ends his alienation and loneliness.