Even though the Greasers and Socs came from radically different backgrounds, both groups faced similar difficulties experienced by most teenagers: finding approval from their parents, searching out acceptance and belonging with their peers, developing relationships with the opposite gender, and entertaining themselves.
Of course, one of the dominant conflicts within the novel comes from the stereotyping and prejudice associated with being from either background; the Greasers and Socs were socially influenced and conditioned by their backgrounds and environment not to like one another. The Greasers had to deal with the damaging reputation that came from being from the 'wrong side of the tracks,' whereas the Socs had to fight the stereotype that everything was easy for them because of their parents' wealth.
The obvious problem that both groups faced was the opposing gang. Hinton set up an obvious man vs. man conflict between the Greasers and Socs, and each group poses a very real problem to the other group. Johnny is beaten to within in an inch of his life and Ponyboy is practically drowned because of the Socs. Of course the Socs are on the receiving end of the conflict at times too as evidenced by Johnny stabbing and killing Bob.
Of course physical conflict is not the only problem faced by both groups. In chapter 2, Cherry Valentine tells Ponyboy that "things are rough all over." What's great about that sequence is that Ponyboy believes her. Through the course of the novel, the reader comes to believe her too. The reader learns that both Greasers and Socs have family struggles and broken familial relationships.
Both groups have problems dealing with emotions as well. I don't want to say that the Greasers and Socs are unique in this either. I think most teenagers struggle with emotions at some point (don't tell my students I said this). In The Outsiders, Hinton shows readers two opposite sides of an emotional struggle. It's actually quite Yin and Yang I think. On one hand, you have the Greasers who wear their emotions on their sleeves. They don't hide anything. They express what they feel. On the opposite end of the spectrum is the Socs. They keep all of their emotions bottled up inside of themselves. As the Socs get better and better at this kind of emotional stoicism, they become deadened to the world around them. I think a lot of the time the Socs fight in order to feel something. . . anything. Once again Cherry shows the reader her deep intelligence, because she points out the above facts to Ponyboy.
"No," Cherry said slowly when I said this. "It's not just money. Part of it is, but not all. You greasers have a different set of values. You're more emotional. We're sophisticated--- cool to the point of not feeling anything. Nothing is real with us. You know, sometimes I'll catch myself talking to a girl-friend, and realize I don't mean half of what I'm saying. . . Rat race is a perfect name for it," she said. "We're always going and going and going, and never asking where. Did you ever hear of having more than you wanted? So that you couldn't want anything else and then started looking for something else to want? It seems like we're always searching for something to satisfy us, and never finding it. Maybe if we could lose our cool. we could."
Ponyboy immediately recognizes what Cherry says as true, because he has seen it first hand.
That was the truth. Socs were always behind a wall of aloofness, careful not to let their real selves show through. I had seen a social-club rumble once. The Socs even fought coldly and practically and impersonally.
"That's why we're separated," I said. "It's not money, it's feeling--- you don't feel anything and we feel too violently."
Other problems that both groups struggle with is a general sense of belonging and social acceptance. That's why they band together in the gangs. It's where they find acceptance and a sense of belonging.