Word of mouth worked for us. Our music teacher has a BA in piano from a music college in Ukraine. She herself is a beautifully talented pianist and she has bonded well with my child.
I would ask for a trail lesson or two before deciding.
Question that haven't been mentioned:
scheduling (do your schedules mesh well?)
vacations (do you pay for them?, can you do a make up?)
missed lessons (do you pay for them? are you permitted to schedule make ups)
do you have a piano that is tuned? If you do not have a piano will they teach a child on a digital piano? (some will not)
billing - some require you prepay a semester, some let you pay weekly, and others monthly.
read the contract - do you have to purchase books directly from them? what is the cancel policy?
Do you want a piano teacher to teach in your home or do you want to go to a studio?
We are a super busy family so we went with a teacher who works for a music school. If we need to cancel and our teacher isn't available for a make up, we can see a different teacher (not ideal, teacher consistency is important), but it is an option.