Ok, so you've decided you want to learn to play an instrument and you've decided to look for a music teacher. What are the type of qualities you should be looking for?
1. Can they play the instrument that you want to learn to a degree of technical proficiency? Well you would think that this would be a given, but there are many people that put themselves up as instrument teachers that cannot play the instrument themselves. There is an episode of The Simpsons where Marge is considering music teaching as an income stream and she says something like "how hard can it be, you've just got to stay one lesson ahead of the kid". Its more prevalent than you think.
2. Performance experience in the instrument being taught. Somebody with a bunch of gigs under their belt may be in a better position to help students with performance anxiety issues.
3. Can the teacher play the instrument in the style you wish to learn? You may be able to learn how to play heavy metal really well from the jazz or classical guy, on the other hand you may not.
4. Instrumental or teaching qualifications. While these do not a good teacher make, they can act as indication of the teacher's dedication to their craft. [sidenote: you may hear somebody explaining that they are a "qualified Simply Music Teacher". Simply music is a teaching franchise, if you pay them a licensing fee for using their programme (used to be $600) then you become a "qualified Simply Music Teacher". Not really a qualification in my opinion]
5. Enthusiasm. Many musicians (and others) teach music as a revenue stream only and have no real interest or passion in passing on the skills. Enthusiastic teachers will encourage you to network with different players, listen to different music and they will tailor their approach to help you as much as possible.
Sure, there are other things to consider I know. All I'm saying is just don't go looking for the cheapest teacher out there, you may not be getting the most appropriate help.