When looking for a teacher, there are several aspects to consider, including (but not limited to) location, rate, credentials, music school vs. independent teacher, attendance policies, and availability.
Location: You want to choose a teacher in a location that is convenient enough that getting there every week will not be a miserable experience, but it’s not a great idea to pick a teacher on location alone. The piano teacher at the end of your street may be very convenient, but make sure his/her training approach, policies and personality are also a good fit.
Rate: You want to take rate into consideration, but making it your only criteria may not be an optimal approach since often you get what you pay for, so going for the lowest rate you can find may land you with some problems. Conversely, the most expensive teacher is not necessarily the best, or at least, not the best fit. Generally speaking rates are set using a combination of factors including the going rate in your area, years of experience, and level of training. If you want a very low rate, chances are, you are going to have to be content with a fairly new teacher, or a high level student. This is absolutely fine, if that’s what you are looking for.
If music lessons would genuinely cause financial difficulty, some schools, and even some independent teachers, do have financial aid and/or scholarships. I've seen some teachers doing a sort of "work study" option with odd jobs for their older students, and some will offer a discount if the family is having financial difficulty AND the student is very dedicated, so by all means, look into those options, just be ready to provide some documentation.
What you should NOT do is contact one of the more expensive teachers in your area and try to get them to give you lessons for the same rate the high school student down the street is charging just so you can feel like you got a bargain. I guarantee that higher rate is there because the more experienced teacher puts a lot more into lessons, not that the high school student wouldn’t be a conscientious and committed teacher as well, but I started teaching when I was 17 and I promise you the quality of my teaching is so much better now it’s not even comparable to what I did then, though I’ve always taken it seriously. The resources I’m aware of, the years of training I’ve done, the years of experience I now have, and the amount of thought and prep I now know how to put into lessons is worlds away from where I started.
Credentials: This can come down to personal preference, and every choice has pros and cons
Like I said above, I’ve been teaching since I was 17. At that point I had many years of lessons under my belt and was playing at approximately RCM level 9. The pros were that my rates were cheap reflecting my (lack of) experience, I definitely had things to teach beginners, had good technique that I could impart to them, knew exactly how the instrument should be set up, and (this is a pro to some) I don’t remember caring if student’s cancelled last minute.
The cons, however, were that I didn’t know how to spot things that would later turn into problems and nip them in the bud, I didn’t have a good way to break down and introduce a lot of things in an accessible, and my interpersonal skills were that of a teenager, so not always stellar.
It was a good introduction to the instrument for my beginner students, and my one slightly more experienced student was really just looking for some pointers and a bit of tutoring on the instrument, so I was able to be helpful to her as well. However, after a B. Mus degree, Suzuki Teacher Training courses, and many years of experience, the quality of my teaching is worlds better and my students now definitely get a better start.
Music School vs. Independent Teacher. Some music schools are very good, hiring only teachers with at least a B. Mus. degree, vetting the credentials of their faculty thoroughly, and providing an excellent musical education. Others are not great, and have a very corporate approach that feels to me sort of like the factory farming of music education. It’s hard to tell based on lesson-rate in these cases, because often the lesson rate at these schools will be comparable to the going rates in the area, however, be aware that the teacher is most likely getting half or less of what you are paying. I do understand why, as any school has overhead costs that they have to take care of, but most of the time, experienced teachers will strike out on their own as soon as they can, or go to one of the higher level music schools.
Always be sure to check out the specifics of any school, and always check out the credentials of a specific teacher you are considering studying with. In terms of schools, I know of a few small schools where the owners also teach and are quite involved musicians themselves, and in that case, I think you can be reasonably sure of the quality.
For vetting a specific teacher there are a number of factors you should consider. Do they have a degree and if so, from which University (or College if you are in the U.S). Does the Faculty of Music where they studied have a good reputation? Do they have any teaching credentials with organizations like the Suzuki Association of the Americas, the Royal Conservatory, or Ontario Music Teachers Association (that last one is obviously region specific). How long have they been teaching? Have they sent students for exams?
Keep in mind teachers also vet their students however, and not every teacher will accept every student.
I feel I should mention that all of these are generalizations and there are exceptions. You could find an amazing teacher with few or none of the mentioned credentials, and you can find teachers who have every credential available but aren’t great, or, perhaps just isn’t the right fit.
Attendance Policy: Most teachers require students to attend lessons once per week. There are many reasons for this, but a major one is that if lessons are sporadic, so is progress, and that leads a student to become bored and frustrated, and potentially have a bad experience with music that could have been avoided with a bit more structure.
It is incredibly easy to start missing lessons regularly without even realizing what's happening if you aren’t careful and don’t make it a priority. One week the student is sick, the next week there’s a birthday for them or a sibling, then a parent has to work late, then there is a school function, and the list is endless. Before you know it, the student is missing 1-2 lessons a month, he or she has been trying to learn Twinkle Twinkle for an entire semester, and now they want to quit because it feels like they are stuck.
Then there’s what I refer to as “the 2 way street”. If a student wants to know that I will always (barring illness or injury) be available for them at their lesson time, then they have to commit to being available as well. I can’t reserve a spot in my schedule for a student who is going to be cancelling regularly (even once a month is a quarter of your lessons). If you cancel that spot, I can’t re-book that time with someone else, as, unlike (for example) massages, or haircuts, almost nobody is looking for a one time last minute lesson.
If you are looking for a less structured lesson schedule (and I don’t recommend that) this again is where you might want to try lessons with a high school student who perhaps might be willing to teach on a less structured basis, as their own school commitments might mean they also have to cancel on a semi-regular basis, and then you’re back to “the 2 way street”. However, talk to them about their cancelation policy, and be up front about it if you don’t want a commitment every week, because the bottom line is that you are entering into a commitment on both sides. Also keep in mind that if you regularly cancel and they regularly cancel, you may end up with 1 lesson a month, which is not ideal.
Availability: Sometimes you’ll find a teacher you think is great in every way…except they only have a couple of openings, and they are not ideal for you. You have 3 choices.
1) Ask if you can be put on a waiting list.
2) decide you will make lessons a priority and find a way to make the scheduling options work.
3) try and find a different teacher that you also like.
What you should not do is say you want the inconvenient spot, then cancel most of your lessons because you can’t make. Yes, I’ve had this happen. It was a spot right after school, I knew it wasn’t great for the student, but he didn’t have any other days free and they said they wanted the spot. Over about 6 weeks he had 2 lessons.
You could have found the number 1 best teacher in the universe, but if you can’t make it to see them, they aren’t going to be able to help you. At that point, you’d be better off with a mediocre teacher you can actually see on a regular basis.