Arguably the largest conflict in the story is that of class conflict, the differences and resulting conflict between the "Socs" and the "Greasers." The wealthier kids have a seemingly different set of priorities than the working-class kids and with their different dress and different mannerisms, etc., they find it difficult to identify with each other and so are in constant conflict. Even when Pony Boy and Cherry find that they care for each other and enjoy spending time together, their differences remain, she can't talk to him when they are in mixed company, etc.
There are other conflicts centered around typical teenage life. The questions of identity that come up in any novel exploring teenage life are explored here too. Is it ok to be a "smart" guy, especially in the Greaser crowd? Pony struggles to find a place for his intellectual interests with his friends given that it isn't typically seen as "cool" or even normal in his group of friends. He feels he has to display a certain amount of manliness and toughness even though it isn't as natural as it seems for other members of the gang.
There are also conflicts around society and how certain people fit in or don't. Pony boy discusses how he used to enjoy going to church but quit going because he wasn't able to dress like the other parishioners and then acted up because he felt out of place. The same thing troubles many of the Greasers as they can't match up with the Socs socially or with their dress so they have to do something to cover up for that frustrated desire.