At the beginning of Ray Bradbury's Fahrenheit 451, Montag's goals are simply to go to work and make enough money to buy the things his wife likes. He enjoys his work:
It was a pleasure to burn.
It was a special pleasure to see things eaten, to see things blackened and changed.
Montag operates like a robot: he never questions what he does, he never wonders about the owners of the homes and books he burns, and he never really notices the world around him.
When Clarisse McClellan enters his life, her questions change his goals slightly. At this point of the story, his goal changes to better understanding himself and his purpose in life—how he fits into this world of repressed thought and speech. Clarisse asks him...
Are you happy?"
Montag laughs it off, at first certain that he is. Soon, however, he realizes he is not. Now his goal is to discover more about himself and the world—we can infer, so that he can find happiness.
When the firemen burn the house of the woman at 11 No. Elm, Montag's goal changes again. It becomes personal for him. He realizes he can never burn another house again.
How can I go at this new assignment, how can I go on burning things? I can't go in this place.
He understands that he no longer fits in his world: he no longer knows his place.
Finally, when Beatty burns Montag's house and tries to arrest him, threatening Faber as well, Montag kills Beatty, as well as the Mechanical Hound. His goal here is to escape from the authorities as they launch a manhunt to capture Montag.
When Montag finally makes it to the river, this begins a new part of his life, where his goal is to join others who want to save books and rebuild society free of the restrictions of the government that has just destroyed itself. Granger tells Montag:
And when they ask us what we're doing, you can say, We're remembering. That's where we'll win out in the long run.
Montag's goals change throughout the story as Montag himself changes.