When people ask me, “What math should I study so I can (fill in the blank)”, the answer I give them isn’t quite what they expect. The best answer to this question is: whichever mathematics you think is the most fun and interesting. This answer doesn’t depend at all on what (blank) is. It doesn’t matter if you want to become a theoretical physicist, a fighter pilot, a computer programmer or even a mathematician, the math you should study is whatever math you enjoy most. That’s because what the number-cruncher ought to learn isn’t any particular subject or theorem, but rather she should train her mathematical maturity.
Mathematical Maturity is what allows maths to make sense. It’s a kind of inner spirit or muse which smolders within every one of us. Some peoples’ mathematical maturity is stronger than others. The way you train it is simple: you do math. Like weight training, to train your mathematurity you need to struggle with concepts and exercises which really challenge you. Material which might have been just right for you a year ago, might today be routine and you won’t grow much from it. Again like weight training, it doesn’t matter how strong you are today, all that matters is that you hit the gym and lift what you can. When I first went to pump iron, I could barely lift the naked bar (45 lbs) with no plates on it; when I first got interested in geometry and algebra, I could barely solve linear equations. Like I could have thrown down that bar and resigned myself to a life as a wimp, likewise I could’ve thrown down the mathbook and consigned myself to innumeracy. The latter course would have been just as silly as the former. Because I pushed through and slugged it out with those equations through thick and thin, today I’m a raging trigonometric Hulk, and difficult exercises tremble at the sight of my shadow 
“But, but, my major requires I take Advanced (fill in the blank) to graduate!” It’s not relevant. Trying to take a specific mathcourse is futile if you don’t have the matheturity you need for it. It would be like a newbie powerlifter saying, “I need to benchpress 250lbs to win the competition for my weight class, I can’t waste time benching these puny 135lb plates!”
How do you tell which mathematics is best to study, to train yourself for bigger things? Fortunately, we have the natural ability to distinguish that material best suited for our perusal. Namely: when you’re ready for a particular area of math, it’ll be fun and interesting and exciting to you. If you’re not ready, it’ll seem like the most excruciating, boring, and/or difficult thing in the world. Therefore, regardless of what your ultimate motive is to learn, you should start with whatever excites your passions. Don’t worry: as you mature in your mathematical sophistication, a figurative blindfold will fall from your eyes, and you’ll see the sublimest beauty and feel the most profound excitement in subjects that were once an opaque soup of gibberish.
The ideal path varies from person to person, and it almost never correlates very closely with the “traditional” ordering of the subjects. Take the branches of game theory and chaos theory, for instance. At a university, these will be senior level courses at least, and yet I’ve known people who were struggling with basic algebra who were absolutely fascinated by these theories. To those people, I say: burn the algebra books and devour the pagan doctrines of chaos! As long as it’s what sparks your interest and keeps you turning the pages long after bedtime. Before I officially got into mathematics, I read books for laymen about theoretical physics. Wormholes, higher dimensions, cosmology, relativity… these things were lightyears ahead of me in the traditional sense, but they’re exactly what I was meant to read about at the time (of course, this is where the weight-lifting analogy fails: weights have a natural order which doesn’t vary from person to person, but one man might be best-suited to begin his studies with fractals while another should begin his with computer programming).
The cool thing is that when you’ve got the appropriate level of mathematical maturity trained up for a subject, it becomes incredibly easy. If you’ve done your training and you’re ready for the literature, you’ll find yourself guessing what the author’s going to say before you even read the words. It’s almost like knowing some close-held secret of the universe, like you and God are winking at each other about your private little in-joke while the rest of the lecture hall sweats bullets over a midterm.
Another important thing is that to attain the desired levels of pure sophistication, you must have a deep, emotional-level belief that you’re good at math. Forget about what any teachers have told you in the past, it doesn’t matter. Past reality is irrelevant to present moment beliefs. Repeat: “I am good at math” until it’s hypnotized into your very subconscious. If you’re busy doing ten things at once and suddenly you see an equation somewhere, your automatic, knee-jerk reaction should be: “Oh, equations, I’m good at those”. That’s how deep you should program the belief in yourself. It doesn’t matter if you have to write it a hundred times a day on paper, it’s worth the effort a thousandfold. Neither does it matter what the objective truth was in the past— the past is history, and in the present, you’re muthaf’ing Good Will Hunting!
FURTHER READING
“Problems” in Mathematics
Five Ways to be Better At Math
Rote Memorization vs. Understanding
Teaching Myself Calculus