Fahrenheit 451 is a wonderful book, and anything I can do to help you better understand it would be my pleasure.
Your question deals with motivation, which in essences asks "why do the people in the book do what they do." I take it that you are asking from a general standpoint, and not in relation to any particular part of the book. As such, let's take a look.
- Montag -- What motivates this guy? That's a difficult question to answer because Montag is the character who undergoes the most changes in the book. What motivates him at the beginning, middle, and end changes. Generally, if I had to put my opinion in, I would say that Montag is motivated by "truth." His character has an inquiring mind, even if it is slow to get moving, and his actions in the story seem to be based around "getting to the bottom" of why books are banned and whether the life they are living is a proper one. I like this as a quote about Montag: "You're not like the others. I've seen a few; I know. When I talk, you look at me. When I said something about the moon, you looked at the moon, last night. The others would never do that. The others would walk off and leave me talking. Or threaten me. No one has time any more for anyone else. You're one of the few who put up with me. That's why I think it's so strange you're a fireman, it just doesn't seem right for you, somehow."
- Clarisse -- Clarisse would appear to be motivated by beauty, creativity, and an inquisitive nature. She desires freedom of thought and expression and her lifestyle revolves around that. As a quote about her, this is a good one: "She didn't want to know how a thing was done, but why. That can be embarrassing. You ask Why to a lot of things and you wind up very unhappy indeed, if you keep at it. The poor girl's better off dead."
- Mildred -- The character of Mildred is motivated to maintain the "status quo." She likes the life she has (at least that is what she says...her subconscious may be saying otherwise) and doesn't like her husband rocking the boat. At least on the dominant level, she wants things to keep going the way they are. Here is a good quote for her...it isn't about her, but it is something she says that sums up her character pretty well: "She's nothing to me; she shouldn't have had books. It was her responsibility, she should have thought of that. I hate her. She's got you going and next thing you know we'll be out, no house, no job, nothing."
- Beatty -- Beatty is probably the most complex character in the book. Like Faber, he is "ideologically motivated." He does what he does in the book because he believes it serves a greater cause (in Beatty's case, he feels it is for the good of society.) Here is a quote from him that shows you what he is like: "She didn't want to know how a thing was done, but why. That can be embarrassing. You ask Why to a lot of things and you wind up very unhappy indeed, if you keep at it. The poor girl's better off dead." Incidentally, this quote also tells you more about Clarisse as well.
- Faber -- Faber, like Beatty, is motivated by ideology. In his case, though, it is the idea of preserving knowledge contrary to the cultural norm. He is motivated by the love of knowledge and a feeling that the current social situation is wrong. Unfortunately, he is also afraid: "So few want to be rebels anymore. And out of those few, most, like myself, scare easily."
Like all good characters, this is just a quick sum-up of their motivations.