Questions such as: what are your methods of teaching? Do you use books, method books, or hands on fingering first? Do you leave room for creativity, so the student does not get "stuck" in the confines of "by the book learning"....something to consider, even research! Very interesting..:) Ask about the use of scales, chords, arpeggios, method books, & sheet music. Will there be any fun pieces to focus on & hold their interest & keep learning exciting? What will the student learn? (take in) & what will they be able to produce or perform? & by when, how long will it take to pass certain musical milestones, or play a real song all the way through? Don't forget to consider the levels that songs are written in. Even famous classical pieces have a hard version, intermediate and beginner. Consider the level of the student always. Will there be recitals? A chance for students to perform for others and experience the sense of accomplishment that comes with that? Let the teacher know your goals. The goals of a student are important, and if you're a parent it is important that you communicate what your goal as the parent is for the student if he/she is a child. It is important that a child student have the desire to learn and play piano. If goals between parents and young students are the same, results will be smoother due to good communication. Everyone should really be on the same page as far as these goals are concerned. Do you want a more structured style of learning, with emphasis in theory & rhythm, directed toward performing & competition? Or do you prefer the student to play for fun & keep up what they've already learned? Both enjoying the musical experience of course. Here are Just some ideas to go over ... Write them down if you need.