This one's a freebie:
His teachers felt this afternoon that his whole attitude was symbolized by his shrug and his flippantly red carnation flower, and they fell upon him without mercy, his English teacher leading the pack. (1.4)
Yep, Cather tells us straight out that the red carnation symbolizes Paul. So, there seem to be two things going on here.
(1) The carnation. Check out "Character Analysis: Paul" for more on this, but basically, the carnation seems to be a hat tip to Oscar Wilde. After Wilde went on trial for homosexuality in England in 1895, his trademark green carnation became unofficial code for "gay"—or depending on whom you ask, "lover of the arts."
So, whether or not this red carnation is telling us specifically that Paul is gay, it's definitely telling us that he fancies himself an aesthete. It's a symbol for Paul of the life he wants to lead; and it's a symbol for his teachers of his being "hysterically defiant" (1.2). In other words, different.
(2) Red. Sure, we tend to associate "red" with being intense, sexy, and daring—but that's not why we want you to read it that way. Cather actually tells us to read it this way. Check out the description of the red glass pitcher that Paul's sisters make lemonade in:
When the weather was warm, and his father was in a particularly jovial frame of mind, the girls made lemonade, which was always brought out in a red glass pitcher, ornamented with forget-me-nots in blue enamel. This the girls thought very fine, and the neighbors always joked about the suspicious color of the pitcher (1)
Red is both "fine" and "suspicious." In other words, it's better than ordinary—that's what "very fine" means in this context—but there's also something a little off about it. Sure, it makes the day extra special. But it also doesn't sit quite right with the neighbors. It's daring. It's unusual. It's excessive. It doesn't fit in. Sound like someone we know?
And here's the other thing about cut flowers: They're cut down in their prime, and they die. As Paul notices that the carnations in his coat are drooping, he realizes that
all the flowers he had seen in the glass cases that first night must have gone the same way… It was only one splendid breath they had, in spite of their brave mockery… and it was a losing game in the end, it seemed, this revolt against the homilies by which the world is run. (2.62)
This little passage packs a big punch. Here, Paul seems to realize for his very own self that these carnations are symbols of his own life, since his big New York adventure has been one short "revolt against the homilies by which the world is run."
Basically, a "homily" is a sermon. It's a little bit of moralizing instruction, like a lecture about how you need to eat your vegetables, or clean up your room, or do your homework before playing Farmville. In other words, boring, bourgeois, and exactly the opposite of how Paul wants to live his life.
So, translation: The carnations, like Paul, are rebelling against conformity by braving the storm and attempting to be beautiful. The problem is, people who try to live a different kind of life—think of the 27 Club—tend to end up dead.