This is something I've thought a lot about. A few things my favorite photographers have said in interviews come to mind. Rosalind Fox Solomon, when asked what she did to improve her work, said that she doesn't think she's gotten better over the years. She does say she had critiques early on from Lisette Model, which surely helped refine her work, but I often think that the greats in photography, herself included, (and to me, Frank, Winogrand, Friedlander, Arbus, etc), were so honest with themselves and brave from the start that they almost immediately started making great work, then hitting different strides over the course of their careers.
Mark Steinmetz in an interview advises that, if you like Eggleston, go out and photograph like Eggleston (or whoever your hero is), your photos won't come out like his anyway and eventually you will get bored. Through this process hopefully you will learn something about yourself, what interests you and how you see the world.
The technical aspects of photography seem straightforward to learn, like anything else, with repetition and curiosity you will get better. The hard part is making an interesting photograph. I think it's a very rare thing and maybe something that cannot be learned. Or at least I hope it can't, because that would mean that there are rules to what makes an interesting photograph, and it's far more interesting that there aren't any. It's a yes or no thing. Either the photo works or it doesn't.
If I were to teach a photography class I think I would spend a short amount of time on the technical aspects, maybe just enough so the pictures will succeed somewhat as far as exposure goes. Then I would try and force the idea of what makes a "good picture" out of everyones mind. Forget about straight horizons, rule of thirds, getting closer, all of it! I would try and teach how to photograph without thinking, and instead shooting with an empty mind and reacting to the physical sensation you get from the world. Maybe a good assignment would be to shoot 30 rolls in a week.
I've heard Tod Papageorge say that "if your photos aren't good enough you aren't reading enough". That's how I think of improving the non-technical parts of your photography. You have to cultivate yourself and that will be reflected in how you see the world. What you read, the films you watch, the music you listen to, it's all brought back in to your photos.
For those interested, here are some interviews that came to mind:
Rosalind Fox Solomon on Magic Hour podcast (which is very good)
Interview with Mark Steinmetz - Flashes
Mark Steinmetz - LENS
Mark Steinmetz - Normal Vision
Henry Wessel - Seeing without recognizing
Winogrand at Rice University