What are some conflicts in The Outsiders in chapters 1 to 4?

The main conflict is between the greasers and the Socs.

The main conflict is between the two rival gangs.  This conflict results in them constantly fighting each other.  It is a self-perpetuating thing.  The gangs fight because they come from different socioeconomic backgrounds, but they also fight to defend their members or to avenge grievances. 

Pony, Johnny, and Dally are at the movies when Dally behaves coarsely to a Soc girl, Cherry.  Despite this conflict,...

The main conflict is between the greasers and the Socs.

The main conflict is between the two rival gangs.  This conflict results in them constantly fighting each other.  It is a self-perpetuating thing.  The gangs fight because they come from different socioeconomic backgrounds, but they also fight to defend their members or to avenge grievances. 

Pony, Johnny, and Dally are at the movies when Dally behaves coarsely to a Soc girl, Cherry.  Despite this conflict, which of course is partly just Dally’s personality and partly due to the conflict between the greasers and Socs, Cherry comes to like Pony and Johnny.  She shows them that she is not against greasers, she just doesn’t like Dally because he is rude.

Pony also has trouble with his older brother Darry. There seems to be a perpetual conflict between the two of them.  Darry is the only father figure Pony has, because both of his parents died in a car crash.  This means Darry worries a lot, and Pony resents him.

It was my house as much as Darry's, and if he wanted to pretend I wasn't alive, that was just fine with me. He couldn't stop me from living in my own house. (Ch. 3)

Darry hits Pony because he does not know where he has been.  He obviously was worried.  However, the incident upsets Pony.  Pony and Darry are constantly fighting, and Pony feels like Darry does not understand him at all.

Things come to a boil when Johnny and Pony are attacked in the park.  A group of Socs finds them there and Johnny accidentally kills one of them in the fight.  Johnny and Pony have to go on the run then, and end up hiding out in an abandoned church.