My teacher (small class, not one on one) has us working on scales as well as pieces of music. He told us to get a left hand fingering that works for the scale of the piece we want to play, so instead of fretting about which fret to play, we know which finger (so the finger for a particular fret depends on the key).
I suggest that you learn tablature. One of the ladies in my class, who is, like yourself, a pianist, passionately hates tablature, but isit is a good way to learn technique and follow someone else's arrangement. You can also transcribe a piece in tablature as you figure out the fingering. Don't worry about mistakes: your ears will tell you if you get init wrong.
Of course, if you want to play Schoenberg, my advice may not be mushmuch use.