As far as we can tell, only two characters in this story ever do anything: Paul and Paul's father. So let's check out the actions they take:
Paul: stays out late, offends his teachers, steals money, runs away, blows the money on fancy underwear, and then jumps in front of a train.
Paul's dad: forces his son to apologize to his teachers, lectures him about fiscal responsibility, gives him a dime, removes him from school, puts him to work, and tries to keep him away from bad influences.
Based on the actions they take, neither of these characters is someone you really want to invite over for afternoon tea. But it does give us a really clear sense of what Cather is doing. Paul is reckless, desperate, anti-bourgeois (that's a ten-dollar word for middle class), and frankly a little dim.
Paul's dad, in contrast, is domineering, authoritarian, well-meaning, and—it seems to us—a lot dim. Seriously, what did he think would happen if he took away everything Paul loved? Is there a best-case scenario here?
Relax, Shmoopers, because this one's easy. Paul's clothes don't fit; ergo, Paul doesn't fit. Check it out:
Got it? Clothes don't fit him in Pittsburgh, where he feels like he doesn't fit in. In New York, however, they fit him perfectly. He's a born New Yorker.
Paul's red carnation is the major player here. As soon as we see him at the disciplinary hearing with a "red carnation in his buttonhole" (1.1), we know that he is a capital-M Misfit. We also know a lot more about him, like he's probably got some deviant sexuality (Cather's words, not ours) going on, and that he's positioning himself as something of an aesthete.
There's also a "characterization-by-negatives" going on. The two pictures over Paul's bed—and you can bet that he didn't choose these himself—are John Calvin and George Washington. Quick rundown on these two gents:
John Calvin: invented a totally strict line of Protestantism. Believed that you were either saved or damned, and probably damned, and there wasn't a single thing you could do about it. "Calvinism" is pretty much a synonym for strict, severe, and did we mention strict?
George Washington: you know this guy. He founded a nation by felling a cherry tree, or something like that. Basically, he never lied. Oh, and he's a symbol of American capitalism and democracy.
So, you put Calvin and Washington together, and what you get is a guy who is exactly like what Paul's dad wants to be, and exactly the opposite of Paul. For Paul, it'd be like sleeping under a picture of this guy every night.