My other answer aside, I think almost all "beginner mistakes" boil down to one of a few things:

1. Not being mindful of what's in the composition. The uneven horizon falls under this, as does the classic photographer's shadow* in the frame, or a personal pet peeve of mine: chopped-off feet in otherwise full-length portraits. I know that when I look at my earlier work I see elements which I know I didn't consider at the time. Addressing this comes from taking time to review your photographs with a critical eye.
2. Technical inexperience. Exposure, focus, all of the how-to-work-the-gear stuff. And I'm including lighting here too. This all just comes with time and practice, and maybe with reading if you're into that sort of thing.
3. Being too caught up in equipment, and blaming what one has or doesn't have for problems. (This comes out in a whole bunch of ways: brand-name wars, emphasizing technical qualities over photographic merit, dismissing all flash photography as awful, hunting for lens defects rather than hunting for pictures to take, and so on.) This is solved by remembering to focus on the other two things and not worry about this so much. There's nothing wrong with photo gear as a hobby on its own, but it's not as fun as photography itself.

* Click that link, by the way. It's on-topic.