Singing is a journey. No singer is ever "finished." And there are different paths of study to get you there.
There is, of course, self-directed study. Such as what I have mostly done. Reading books and programs and doing what I understand from them. First off, not everyone learns the same way. Some are more visual. You can talk or write about breath management all you want but seeing the motion of a person's body can be the key for many folks. Robert Lunte has probably the best tutorial vid I have ever seen on breath management. Just watching that one vid is worth a whole book of verbiage and imagery.
But that leads to the next avenue of learning. Visual learning. You could read about it. You might "hear" it. But seeing it happen clears everything up. Such as Lunte's video on embouchre. Not exactly how I related to the word. I learned the word from my brother when he studied clarinet. But more to the point, you "see" what Lunte means by embouchre and now you can link it in your mind, in living color, to what the concept is.
And then, auditory. Every person's body is different. But if you can hear the sound and re-create it in your own body by whatever words your mind has come up with, then you have it. I have done this with students when I was teaching electrical trades, which requires a lot of math. A student made up his own word for reciprocal and rather than obstinately insist on the technically accurate word, I let him use his own image. And he made a perfect 100 on his math test, the next day. And there was no "curve" in my grading. Students earned their grades the hard way, by giving the right answer.
And therein lies the advantage of the next path. Having a teacher, especially one sensitive to your needs as a student. And for that, in singing, the teacher has to hear you. Preferrably, in person, in the same room, if possible. Though, these days, locality is no longer a hindrance with the use of Skype and other internet-based real-time communication utilities. Although, I will still prefer in-person. I know a little too much about electronics to ignore totally the artifacts that occur in translating an analog function like the human voice into digital signal and back, again, and including the response curves of both the microphone and the head phones or speakers, at either end. But there have been good results in spite of these limitations. For, in recording, we are not hearing each other acoustically, anyway. And, as others will point out, a good note is a good note, regardless of the quality of equipment being used.
And here is the advantage of having a teacher. Your learning and improvement happen so much faster. And it is not because you build muscle so much faster by spending money. It is because the teacher can spot mis-steps immediately and give you the solutions immediately. The sooner you discard the mis-step and do the correct thing, the sooner you improve,
Just like when I was teaching electrical trades. Could someone eventually learn by reading and trial and error, to bend a 4-point saddle in EMT like I can? Sure. And it will take a while. No, it didn't take me a while but that is beside the point. The point is, by having me physically show a student, he was bending 4-point saddles with professional consistency after about an hour. And directed his own practice by finding things around the shop, obstructions of structure that would require his new-found "skill." Something he could do all along, provided he had the right path. That's not bragging for myself but to show the advantage of having a teacher show you what is right, from the start, before you develope incorrect habits.
And, in fact, it had as much or more to do with the student as it did with my instruction. He had the motivation, the desire to learn, to put aside what he may have previously thought, at least to consider what I was saying.
And so, when seeking a teacher, ideally it will be a teacher who can reflexively modify his lesson plan to suit your needs. Accent in language can be limiting. So, just telling a student sing ah in father doesn't help if the accent of the student is such that father is pronounced with the high a in the word cat. Tell him to say ah and he still sounds like a whiny cat. It takes a flexible teacher to tell the student to roll forward more to aw to kind of over-correct for the backwards a he as been using.
And dipthongs, the bane of american english. Especially in southern accents where there is a tendency to create more syllables in a word. I don't know if it is because of the slower pace of life and we have to elongate conversation to fill up the days, or what! LOL
And accents where too much sound goes through and out the nostrils. Granted, that is maybe more about style. But leaving these "stylings" in place can have a destructive consequence. Which places the onus on the student.