It'd be easy to spend multiple lifetimes studying math, so you'll have to set some priorities. Applied vs. pretty, pragmatic vs. rigorous, discrete vs. continuous, and various subfields within "applied," e.g. So presently, when you have a better idea what your priorities are, you'll probably want to pose a variant of the question again.
(E.g., not "what are the 'canonical' sources for math" but something as specific as "what are the 'canonical' sources for math leading up to what I'd need to understand X" where X is something like "the cryptanalysis of the Data Encryption Standard" or "the proof of Fermat's last theorem [good luck:-]" or "why people think Y's work was important" where Y is Galois or Hilbert or Ramanujan or Noether or Erdos or Matiyasevic or whoever.)
Meanwhile, if you just want to see what the fuss is about before trying to formulate a more specific question, I can recommend any of four kinds of samplers.
1. For about 80-90% of ways of analyzing the physical world, one really wants to know calculus. Get _A Concept of Limits_ (cheap from Dover), the three most promising calculus books from your local library (and/or webbed tutorials), and a basic dealing-with-the-physical-world book which assumes you know calculus (e.g., just about any serious physics text, or _The Art of Electronics_, or something acoustics or signal processing or whatever). Keep fiddling with them, and doing exercises as necessary, 'til the pieces fit together.:-| Expect it to be quite a lot of work --- by my estimate, freshmen and sophomores at Caltech in the 1980s generally spent at least 250 hours on it, sometimes more like 1000. And it will probably be much easier if, like them, you can arrange to get at least 1 hour of feedback every 20 hours of study from someone who already understands the stuff.
2. For anything in computers, getting familiar with the basic math of reasonably serious algorithms is really useful. I, like many people, like _Introduction to Algorithms_. Get it and study it; understand at least a representative number of chapters. My estimate is that this is a lot easier than option #1, maybe five times easier. It isn't anywhere near as big a hammer for dealing with the physical world, but it can be extremely handy for dealing with the software world.
3. If you want to see what all the fuss is about in some representative areas of less-physical, less-computer-y math, I know of two Dover books which try to drag you from advanced high school math to a famous math result. _Abstract Algebra and Solution by Radicals_ drags you through (the modern, cleaned up and rigorous version of) Galois' proof that there is no closed-form formula for solving polynomials of fifth order. _Computability and Unsolvability_ drags you up to Matiyasevic's proof that Hilbert's tenth problem is insoluble. Working through either of them in detail would be a lot of work, almost certainly more than you want to do if your interest turns out to lie in something else like graph theory or algorithms or topology or statistics. But you could probably learn a lot about roughly how things are done merely by skimming either of them a few times. (And if just seeing broadly how things are done is your priority, you might prefer _AAaSbR_, since showing broadly how things are done seems to be one of its priorities too.)
4. Peter Winkler's newish (2004) _Mathematical Puzzles_ book is also very good and very math-y and well worth looking at as a sort of inspiration. However, if you ever get tempted to think that the extreme elegance of puzzle solutions is representative of how math gets done, look back at section 3 before jumping to conclusions.
"I am lost as to where I should start. I want to have a fundamental, intuitive understanding of it."
My closest thing to a literal answer to that would be: read _AAaSbR_. Like it very, very much.:-) Like it so much, in fact, that you are motivated to really study something like _Algebra_ by MacLane and Birkhoff (which is like a big watershed in which _AAaSbR_ is but one stream). After you get your mind around a good chunk of that (enough that you feel no great fear of an open-book exam composed of exercises from your choice of 20% of the chapters, say), do some variant of the calculus stuff I described in section 1 to see how abstract math ties into the stuff people analyze in the physical world. But I doubt in fact this is what you want. I suspect it'd be more than a full-time year of work for most people. And even if you had the time and energy, well before you finished I think you'd probably prefer to stop studying the foundational stuff so deeply and start to climb up some shortcut to some application or subspecialty.
Incidentally, mooneater's advice "algebra [...] Be very comfy with that before proceeding" is good... but note that it's referring to a high school algebra which has rather different priorities from something like what MacLane and Birkhoff mean. Don't try to follow mooneater's advice by going to a university library, taking down a book titled "Algebra," and running away screaming "math is not for me." I learned a lot of useful math, did my Ph. D. on quantum mechanical Monte Carlo simulations, and only understand a little of MacLane and Birkhoff (but have looked parts of it in order to try to understand a little bit about "categories" and some other stuff, and would consider more time spent understanding it to be time well spent).