#1
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?
#2
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.
#3
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.
#4
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.
#5
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.