umbran seems to have hit all the touchy feely points. He's right.
Now for the tactical side. Put the emotions aside (which now might not be the right time) and look at the situation tactically.
The "Let's Be Friends" statement could mean a lot of things. In general, it is best to interpret it as let's not see each other or seek each other out, but if you see each other, you'll be cordial.
It could also mean she wants to keep her hooks in you, blocking you from moving on. If you follow the interpretation above, it will help prevent that. If you keep seeing her, talking to her, inviting contact, you're enabling a negative usage of Let's Be Friends.
Next, an ex, is an ex for a reason. Yes, it is true that some couples break up, and then re-unite, stronger than ever. But those are the exception. Generally, couples break up because there are problems between the 2 parties. If something hasn't dramatically changed, the problem won't go away the second time around.
Generally there's 2 vectors for a relationship ending. One is that there is something seriously wrong with one party (sociopath, abusive, etc). Or that there is a number of compatibility and communication issues between both parties, wherein both are at fault for the failure of the relationship.
You don't have to solve the former type. If she's crazy or evil, it ain't you, and just put a lot of distance between you and her. If it's you, well, you might need some serious help, and if you're that broken, you shouldn't be in a relationship. If she was cheating, you don't want her. A cheating type generally tends to cheat when they're not happy where they're at. If they cheated WITH you, then they are likely to cheat ON you. Stay away from that kind of people.
The latter type is solveable. Don't solve it for her. Solve it for yourself. Think of each thing you had "wrong" with her, and answer the question of "for my part, I could have done XYZ better". For normal people, when there's problems, while one person MAY have started it, both could have acted better.
It's common in a relationship, for one side to want to change the other. That's nonsense of course, you can't change people. At best, people can change themselves, and it can be hard to do. It'd be great to find somebody who loves you for being you. I recommend trying to be the best version of you that you can be. Be healthier, politer, cleaner and any more positive "er" words you can be. It'll have to do.
Additionally, look at your life. Is it balanced? Are you stable. You're young, just starting out. Hopefully in school. If having a food, work, shelter is uncertain, you ain't ready for a relationship. If you have no time in your life for your girl, because you're always working or playing a game, you have an imbalance, and you won't be able to maintain a relationship. Sure, the high maintenance girls want all your time, but even the "normal" girls need regular attention doing normal and special things (special does not have to mean Grandma sensitive activities).
You also want to seek out a girl who is also stable and balanced. If you ride in as a white knight to save her, that relationship only works while she needs a white knight. Once she's stable, the dynamic changes and it can go kaput.
Ultimately, both sides gotta be in a state of fecal coagulation. Otherwise, it's just a mess.