I am looking for a rough guideline of the steps I need to take in order to gain an advanced understanding of the theory and applications of mathematics. What skills would I need to perfect in order to transition from my current (relatively) low level understanding to that of something close to a Master's graduate?
I'm by no means ignorant to the difficulty of this task, I'm fully aware that it will take multiple years to complete. Just simply seeking advice :)
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You don't say exactly what level you're starting at, but let's say you only know elementary arithmetic.
Then the main pre-university subjects would be:
At university level, the central subjects are algebra and analysis, but many other subjects branch out from these. For the undergraduate university level, the schedules from the Cambridge undergraduate course in mathematics will give you a good idea of what there is to study.
See here: https://www-wp.maths.cam.ac.uk/documents/schedules.pdf/
I would recommend that you applied to engineering. I did aircraft engineering, but naval or architecture are also good disciplines.
If you are not ready, then high school textbooks. You can probably do more than 10 pages daily or 100 pages weekly and while you progress, you will also learn do discern between good and bad pieces of information as presented in the textbooks.
While at it, I recommend high school physics as well. Mechanics, dynamics, fluids and electricity. Optics and quantum physics tend to be too difficult and I would skip them.
Mathematics is about developing a sense of right and wrong out of true and false statements and implications. Go slow, use alternative sources, even online courses. Sometimes you will need to study history of mathematics, some other times an internet reference will be enough. If you still don’t get it, mark it as “rework” and get over it. Is important to compile your own list of misunderstandings and questions. A later review will help you understand yourself better, by the way the mathematics eases into your mind. Some obvious answers will surprise you for even asking the question, while some other misunderstandings will follow you. Don’t get stuck because you don’t understand. You need to ask to be answered. Enjoy!