Our first introduction to Mildred is her attempted suicide. This is a very powerful and dramatic way to introduce one of the key players in the novel; it is a great way for the reader to realize, almost immediately, that something is wrong in their society, and in Montag's marriage. So why introduce Clarisse first? She provides the perfect contrast to Mildred. If we had immediately been introduced to Mildred, her suicide is awful, yes, but without the contrast of the fresh, youthful, very much alive Clarisse first, it's not as powerful of an impact.
Compare the descriptions to the two women, and how contrasting they are. Clarisse: "eyes so dark and shining and alive that he felt he had seaid something quite wonderful," and "her face...as fragile milk crystal with a soft and constant light in it," and "her eyes were tow miraculous bits of violet amber that might capture and hold him intact." Then, Mildred: "a body displayed on the lid of a tomb, her eyes fixed to the ceiling...immovable," and "her face...felt no rain...felt no shadow...her eyes all glass...not caring whether [breath] came or went, went or came."
If you compare these descriptions, the difference is stark, startling even, but even more so because Clarisse's came first. She is so alive and dynamic, and Mildred is like a corpse. It's an effective way to show society's impact on people; Mildred is the result of their society, Clarisse the potential without it.
The introduction of Clarisse, who is atypical of women in Montag's society, forms a clear contrast with Mildred, a woman whose who life revolves around what she is told by her TV "family". Clarisse is used to thinking for herself. She has read books and asks Montag a critical question, "Are you happy?" The scene quickly changes as Montag enters his house only to find Mildred glued to the TV and complaining about not having an extra TV screen. In one short moment, the audience is able to identify the differences and the negative aspects between the society that Clarisse envisions and the real society which Mildred represents. The rest of the novel is then able to examine Montag's pull between the two ideals of society and eventually choice to reject the society that Mildred represents.