The Right Teacher... eeny, meeny, mynee, moe?

Remember, how diligent you are in the selection process will determine just how successful this adventure will be.
Get a recommendation for 2 or 3 names. Call them and have a general chat on the phone. If you like their attitude, book a lesson.

See how it goes. Stay for the lesson. If the relationship works, book a schedule. If it doesn't, try another teacher.
Understand that this relationship will make or break your child's interest and enthusiasm.

As for myself,

I am very insistent on the relationship to be a good fit. I will recommend another teacher if I feel it is not a fit.
I also insist on the parent staying for the first few lessons to get a feel for how I do things.