Clarisse is introduced prior to Mildred to create contrast, for all three characters (Clarisse, Mildred, and Montag) are different.
To one extreme you have Clarisse, who is a free thinker, which in this society, will lead to her demise. She is also young and innocent. On the opposite extreme you have Mildred, who is addicted to the empty, vicarious existence offered by the shallow popular culture pumped in through the TV walls. In between the two extremes is Montag.
From Montag's brief encounter with Clarisse, we learn a lot about him. We learn that he is thirty, a fireman, and a figure people often fear. There is a contrast between Montag and Clarisse. Though she is seventeen, she has a child-like quality about her. She is open to the natural world around her and a bit of a risk taker. Montag most certainly is not. He has his routine and is blindly following along, rarely stopping to take in the wonder in the world around him (in another year or two, would he be in the same empty, hopeless place as Mildred?). She poses the question he has stopped asking himself, a question that begins to shake him free of his routine, "Are you happy?"
It is this question, and its eventually effects, that begin to push Montag away from becoming a blind follower like Mildred to a free thinker who questions everything, like Clarisse.