Now that you know what your goals are in terms of music lessons, here are some very general pieces of information which will help you to make an informed decision regarding your child's potential music teacher.
Education
Typically, education is the first thing parents want to find out. It is the shortest means of discovering any kind of credentials. In the past, I have not stressed a formal education due to my notion that instructors and musicians could be very effective through experience alone. Of course, it goes without saying that this is still the case – I myself began lessons with an older woman teaching out of her home, she never attended university, but she had been teaching for about 50 years, and simply by virtue of that fact she knew the needs of her students very well. On the other hand; a college education does create the opportunity to attend classes such as pedagogy and child/adolescent development courses all the while honing one's instrumental ability.
Literature
This is an area that I find profoundly important, much more so than music teachers having a four year (or more) degree. An instructor must be continually learning in order to know what is out there in terms of the teaching world. Magazines such as “American Music Teacher,” “MENC Periodicals,” and “Music Teacher Magazine” are some examples of publications that contain wonderful information on practical instruction.
In addition to magazines, a good instructor should have several “go-to” texts which forms their theoretical basis in the studio. Most major areas of study have some seminal book to its instruction, you could ask in the interview or in conversation to which books your child's potential teacher adheres. If she/he is able to converse with you and explain what they liked about the book, its a good sign that they pay attention to what other experts have to say regarding music instruction.
Which leads me to the next point...
Are They in Lessons Themselves?
Many parents/students seem surprised when an instructor states that they are still taking lessons. It shouldn't really be shock, continuing personal and professional development shows dedication to the craft – something that is very admirable in any expert. This is something that is absolutely vital for a teacher maintaining a realistic vantage point during their own teaching. An instructor enrolled in lessons sets a good example to all of that instructor's students – it also give the instructor another expert with whom to discuss issues that arise in teaching. It should be noted that for a professional musician, weekly lessons might not be necessary, but at the very least, a teacher should check in with another instructor ever couple of months.
Professional Memberships?
A good instructor will have a membership in at least one professional organization. Music Teachers' National Association (MTNA), National Association of Teachers of Singing (NATS), and North American Music Teachers' Association (NAMTA) are just to name a few. Professional organizations provide a network for teachers to collaborate, share ideas/knowledge, recommend products and a host of other benefits. Most professional organizations also have yearly conferences where experts present, new literature is shown and technology unveiled all with the hopes of making private studios better.
Background Check?
Most studios have some sort of background check procedure. Many opt for the state fingerprint card which is bar-none the most reliable. There are however many private background check companies that do a comparable job in looking at criminal records as well as enquiring about general reputation and searching the internet for information. NAMTA members have an option to submit to a background check which shows up on their teacher profile, other organizations might have similar services. In the end, it is up to you how important this aspect of the selection process is.