aeh's post is really great. I'd say though that a keyboard with weighted keys is totally appropriate for a beginner. I'm a mid-level pianist (good enough I can teach beginners, not good enough to go pro) and I find the keyboards with weighted keys comparable to decent uprights and better than junk pianos (not least because they don't have to be TUNED, which most families I know with real pianos don't do often enough).
It's hard to vet teachers! I went with a teacher for my daughter (since she doesn't really listen to me) who fills some of our needs -- she is really good at getting DD to enjoy playing, and connects with her on a warm level, which at the time DD needed more than anything else. But she teaches almost no technique to her, which I didn't realize until we were a couple of months in (I wish I'd been able to see a recital, but we joined this teacher right after she'd had her recital). We stayed with her another semester because of her warmth, but I'm going to try to find another teacher in the summer.
a sense that more advanced students are actually more musically and technically skilled, and not just playing more notes.
Ugh, totally this. The teacher we're with turns out students who... can play more notes, at the end of the day, which I realized when we finally saw a recital.
