Six of my students just moved out of town, one an hour away, a brother and sister to Iowa, and three siblings to Nebraska. All of them wanted to know how to find a new teacher.
Here are some questions I suggested they ask of potential teachers:
*What are your qualifications?
(I would look for a teacher who also performs publicly, at least sometimes.)
*How many students do you have?
(Personally, I'd look for someone who has a lot but not a huge amount. This shows they're established but will still have time for your child.)
*Do your students perform publicly? If so, what kind of opportunities do you offer?
*What kind of music do you require your students to learn?
*Do you teach playing by ear as well as simply reading notes?
*Do you teach composing or improvising?
*Do you teach your students how to use lead (chord) sheets?
*What are your studio policies?
Asking these questions can give students and their parents insight into what they could expect before they commit to lessons.