[ZenGum]

Well.... you've got a possible. It might develop, or it might not.
Go shopping. Before the shopping trip, observe what other "cool" looking people in your society wear, so you've got an idea of what you're aiming at.
Go shopping. Dress fairly nicely - this is a proto-date. Take money. Don't be stingy (but keep your head, of course). Listen to her advice. As a general rule, at that age females have much better dress sense that males. Don't spend too long deciding. That is irritating for her. Keep an eye out for things she might be interested in. Understand that she will probably "just pop in for a minute" to a few shops of her style. Wait patiently.
Then once you've bought your new glad rags, be sure to insist on buying her a coffee (and cake) to "thank her" for her advice. Then, be sure to arrange that you're going to the dance with her.

In other words, do the exact bloody opposite of everything I ever did when I was a stupid inept teenager without a bloody clue.

The path above does not ensure success, but the alternative pretty much promises failure.

If nothing else, you'll have (a) a nice new outfit and (b) probably someone will notice you out with a girl .. which subtly improves your overall social standing.