I was in a relationship with this girl, it was my very first and we got as far as holding hands. We broke up because she felt I've kept things from her and, you.
When she told me she wanted to break, I nearly broke down and literally cried..
I couldn't concentrate in school anymore. Had a sudden obsession with The Beatles. (latter not necessary I guess)
Simply, she told me she doesn't hate me. And I've tried ways to get her back, but she keeps pushing me away (not rejecting). But recently I found out she cried over another guy because she felt she bought him harm (from her, she treats this guy as a brother, he comforts her when she's down and when we broke up) and I'm comforting her now.

What should I do? Even if I do nothing, I will always think of the times we've had. How?

Leave her alone. I suggest you back up and focus on yourself. Check out the no contact threads in my signature.

She already made her feelings about you very clear. She wants to be left alone. Forcing the issue isn't going to help either.

Work on yourself while you recover from this rejection. Maybe one day you can be together, maybe never. Who knows? But right now, it's not the time to be with her so leave her be.