What are the contrasting characteristics between the different teachers in the book?

In To Sir, with Love the staffroom (what we in the US would call the faculty room) is a central place from which Mr. Braithwaite, the narrator, is able to observe and evaluate the other teachers and the different aspects of the teaching world and of society they represent. The teacher who comes off worst is Weston, a cynical and angry man who does not seem able to say anything that is not sarcastic or critical. He makes racial jokes which show both his insensitivity and cluelessness. In today's world, such a person would be sacked his first day on the job, and deservedly so.
On the other hand, the women, especially Miss Blanchard, are shown to be generally sensitive and positive characters in their sympathetic attitude to Braithwaite and to the students. Mrs. Drew, we are told, displays a "gravity behind her remarks, indicative of a really deep concern for the children." Miss Clintridge and Mrs. Dale-Evans are both East Londoners themselves and because of this have an inner and more reflexive understanding of the students. Miss Dawes and Miss Phillips have bonded with each other and formed what Braithwaite calls a "tight, secret conclave of their own." Dawes is called a "professional virgin" to her face by Weston, another indication of how radically things have changed since To Sir, with Love was written. Other characters in the novel make remarks either directly to Braithwaite, or loudly enough that he can hear them, showing that although Britain did not have the legalized discrimination that much of the US still had in the 1950's, when the story takes place, the attitudes towards black people were just as prejudicial as in the US of that period, in many cases.
In general however, the teachers at Greenslade are, with the notable exception of Weston, nice people, to put it simply. Braithwaite's breakthrough is not simply his ability to win the students over in spite of his "outsider" status, but to create a more progressive approach to teaching than even the headmaster, Florian, has already initiated. It is not, in the novel, quite the dramatic thing occurring in the movie version in which Braithwaite (Sidney Poitier) drops a stack of books into the rubbish bin and declares, "These are useless to you," but a more subtle approach in which he tries to relate the curriculum to the everyday working-class world of the students. Perhaps because Braithwaite comes to an almost entirely white class as one of a different race, he is paradoxically able to accomplish things in teaching which the more conservative teachers are unable or unwilling to do.
For American readers, it's usual to draw comparisons between Braithwaite's book and the other famous novel of the same period told from a teacher's point of view, Evan Hunter's The Blackboard Jungle. To Sir, with Love is unsurprisingly a relatively gentle, positive story. Hunter's novel is far more cynical and violent, perhaps in a sensationalized manner for its time. Both the differences between the US and the British school world of the time (and society in general) and the similarities are made clear by a reading of both of these novels.