If you are 'professional' guitar teacher, how did you go about developing your relationship with the music store? I've asked a handful of teachers I know, and they all had a personal connection (friend, family) with the owner or manager of the music store. In one exception, the guy became a teacher after being a long-time student.

Has anyone had success using a more traditional approach? (I.E. just showing up and asking to speak to a manager, and explaining that you'd like to teach) Is it totally unorthodox for me to ask for a job submit a resume out of the blue?

How did you end up teaching guitar at your music store?
I was a guitar teacher for extra cash throughout college. I never worked in a store, but the guitar shop I work in does have lessons.

The best thing you can do is to already have clients. If you're renting the store's space (as is common with lots of store/teacher relationships) they'll want to know that you can fill your time. This works well for you, because if you can fill your own to an extent and already have a bit of a name, it'll be easier for the employees to speak towards you and help you find more clients.

Also, being a frequenter of the store helps. I would say that you should be relatively well dressed when you come in each time, and be super friendly.

Have a good approach. Don't ask "Are you guys looking for teachers?" say something like "I would really like to teach at your store. How do I go about doing that?"

Also, not being a stuck up douchewad helps. There are a LOT of those. We turn down most teacher applicants. Look like the kind of person a parent would be okay leaving their kid alone in a room with for a half hour at atime.
Thanks for the advice!
You said you turn down a lot of teacher applicants.. what ways do you have people apply?
I'm thinking I should ditch the standard resume, either replace it with a music-teacher specific document or drop it completely. As you said image is obviously super important here.
A resume of where you taught before is absolutely useful. Definitely give one if you have it. Also useful, a registry of your students, and have numbers of ones that are willing to be contacted about you so they can give a reference. It's a job just like any other.
You just go into store with your music focused resume and see if they have any teaching positions available. Be ready to prove you can play if asked. Having a music diploma or degree and teaching experience (even if its not paid) helps a lot as does being able to play very well.

Its that simple. It was for me anyways.
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Unless you're sure she likes you, telling her you like her has a 110% chance of failing.

But hey, at least you have a 10% chance of absolutely guaranteeing failure.