What is the central conflict in "Where Are You Going, Where Have You Been?" by Joyce Carol Oates? When, where, and how does it develop or become...

I am not allowed to write an essay for you, but I believe that I can get you on the right track by answering several of the questions in the prompt.  

There are several conflicts in "Where Are You Going, Where Have You Been?" It's going to be up to you to decide which one you think is most "central."  There is a conflict between Connie and her parents.  She desires to be more independent, and she also sees them as too traditional.  There is also a conflict that exists between Connie and her sister, June.  It's not a huge conflict, and I'm not certain that June is even aware of it.  The conflict is that Connie has a problem with how "plain and steady" June is.  Connie doesn't understand how June doesn't desire to be more flirty and cool.  I don't believe that either of those conflicts are the central conflict.  For the central conflict, I would definitely write about the conflict that develops between Connie and Arnold Friend. 

The question asks about the nature of the conflict and how it changes.  I believe that those two questions come together.  When Connie first sees Friend, there's very little conflict to speak of.  He creepily says "Gonna get you, baby," but Connie just keeps walking.  The encounter seems to be over and done with. 

It's not until a few days later that the conflict between Connie and Friend becomes extremely serious.  Readers are told that Friend shows up at Connie's house near midday on a Sunday.  Connie opens the door to see who it is, and Friend immediately begins trying to convince her to get in his car and go for a ride. 

"You wanta come for a ride?" he said.

Connie smirked and let her hair fall loose over one shoulder.

"Don'tcha like my car? New paint job," he said. "Hey."

"What?"

"You're cute."

At first, Friend's tactics resemble confident flirting; however, as Connie continues to fend off his advances, Friend becomes threatening.  His threats are a combination of psychological and physical manipulation.  He knows things about Connie and her family that he just shouldn't know.  He knows exactly where her family is and how long they are likely to be gone.  

"But my father's coming back. He's coming to get me. I had to wash my hair first—'' She spoke in a dry, rapid voice, hardly raising it for him to hear.

"No, your daddy is not coming and yes, you had to wash your hair and you washed it for me."

Eventually, it becomes clear to Connie that Friend is really bad news.  He threatens to physically come in the house after her if she attempts to call the police.  By this point, Connie is so scared that she decides to risk rushing for the phone.  Friend comes in the house as he threatened to do, and he stabs her "again and again with no tenderness."  She's not being stabbed with a knife or anything like that, and critics have various opinions on what is actually happening to Connie; however, the result is still the same.  Connie leaves the house with Friend, and she is unable to fight any longer.  She's a complete shell either because she's so scared or because she has given up.  

The final part of your question asks about the conflict's resolution.  The conflict is resolved when Friend wins.  Connie leaves her house thinking that she will never see her family again.  

She thought, I'm not going to see my mother again. She thought, I'm not going to sleep in my bed again.

This story is one of those rare times that the "bad guy" wins.  In my opinion, the resolution is not satisfying.  Perhaps it is more realistic than the ever-present happy ending conflict resolution, but happy endings are satisfying to me.  I love teaching this story to classes because students are upset at the ending.  They are not happy that the story doesn't have a clean, happy, and satisfying conflict resolution.