I don't see how one section could possibly be more important than the other; they all work together, hand-in-hand, to develop the entire storyline. The first section, The Hearth and the Salamander, establishes the main characters, the background exposition for the setting and the society, and lays the foundation for what is to come next. We meet Montag, who on the surface appears to be happy, but soon realizes that he is miserable. His misery sets him in a quest for understanding. We meet Mildred, who also appears to be happy, but through her suicide attempt, we know that she isn't. We also meet Clarisse, who is truly happy. Clarisse is an essential catalyst in Montag's own journey of self-discovery and enlightenment. All of these elements are crucial for building to the next section. The themes of shallowness, oppression through entertainment and happiness through individuality are all introduced.
In The Sieve and the Sand, Montag develops dramatically as a character, whereas most of the other characters remain static. Mildred doesn't change. Clarisse, unfortunately, is gone. We meet Faber and together with Montag, they plan their rebellion. For the first time, Montag feels excited and happy, and isn't as confused anymore. He starts to lash out against the system. Bradbury's themes of ignorance are developed through Mildred's friends. We see a turning point in Montag's life and character.
The last section, Burning Bright, is where everything that has been developing to this point comes to a head. Montag makes crucial decisions that change his life for good. With the murder of Beatty and his disillusionment with his society, he consciously makes the choice to be an outcast. The bombs and destruction of his city are secondary to the evolution of Montag's character, which completes itself in this section. He has gone from a confused, frustrated, blind character to an empowered, driven and decisive one. He understands the nature of his society now, and is willing to sacrifice all to change it.
So, all work together, hand-in-hand, to fully develop the characters and themes. I hope that helped; good luck!