Training
Ideally, an Instructor has been formally trained in a music school so that they have the extensive knowledge needed to teach the wide field of music and styles to their students.
So you may ask an Instructor what their educational background is.
Performing Experience
A musician should have several years of performing experience to have familiarity with various musical styles, key artists, specific popular songs, etc. This could be in school bands, performing solo or in a combo, anything that gives practical experience. What is the teacher's performing background?
Teaching Experience and Ability
Teachers vary in their skills in regards to how they plan a program for each student, how they explain each musical idea patiently and how they get the student accomplished enough with each step before venturing onto the next level. Unfortunately, aside from the breakthrough subject of "Study Technology", there is very little available in schools today for actual one on one, successful teaching techniques leaving instructors to develop their own approach.
I'd suggest asking, but few teachers have a method at all and would be surprised at the question as they "just teach". If you don't ask, just see how things go after a few lessons.
Teacher Personality
And now, let's put in a plug for finding a teacher with that winning, friendly personality who is a pleasure just to be with even before the main event starts. A positive, encouraging, yet not overbearing attitude goes a long way in keeping a student of any age progressing while enjoying his musical adventure!
No need to ask, just see how you feel when you talk to the teacher!
Teaches Real Music, Not Shortcuts
Some teachers are known to skip actual music education. Instead, they jump right to songs using simple and barely educational charts to please beginners who none the less can't really play the tunes yet. After a while, the parent or student notices that they are not progressing or understanding and generally seek a new teacher. The ideal scenario is to start the student on a series of good, educational instruction books and then add various exercises and their favorite songs as skills permit.
Definitely ask an Instructor what their lessons plans are like.
Price
Price is actually not a criterion for selecting a teacher. In the first place, most fees are in the same range within a couple dollars. Therefore, the most affordable teacher will be the one who makes the most progress with the student week by week, dollar by dollar. Just look back at your progress the last couple months and you'll know what you're getting for your hard earned $.
So, make "How much do you charge?" your last question after asking all the other questions!
The Studio
Store or school studios tend to be small, bare bones, shared rooms, often in a basement, with no windows and few accessories beyond two chairs, a music stand and maybe a boom box.
In comparison, the private instructor can design a special room for lessons which would be a larger room (big enough for a parent to sit in), a good sound system, extra books, folders filled with songs and other music charts, extra guitars and amplifiers, pictures, plants, etc. Overall - a more functional and pleasant atmosphere and working environment.
Ask the instructor what the room is like. Is there an amplifier? A sound system? Can I bring a CD or iPod? Is there a place to sit for a parent? Etc.
There you have it. A few things to keep in mind when searching for a music instructor to ensure the best results and the most fun!