> You can take a general function and break it into its 'component' polynomial terms, this is called the taylor series.

> Fourier transform is essentially similar to the fourier series except you're allowed to have non-integer frequency multiples.

Not really. Taylor series are "optimal" in a completely different way from Fourier series. Since the "decomposition strategy" used by Fourier series is completely general (there are polynomial equivalents of the Fourier Series) it's worth drawing the distinction.

The Taylor Series gets all of its information from the neighborhood of a single point. Thought experiment: consider the curve e^x. Now set it to 0 everywhere outside of the interval (-e,e) where e is arbitrarily small. Now take the taylor series of the curve at the point x=0. You get back the original curve e^x -- the Taylor Series completely ignores the fact that your curve no longer looks anything like e^x! You can do even worse than that: paste a tiny, almost-linear segment of 1+sin(x) into e^x around the origin. You couldn't tell the difference looking at the curve (they both look like e^x) but the Taylor series would converge to the sin, not to the e^x curve you "transplanted" the patch into. Taylor series are good for answering the question "How does f(x) change if I nudge x a bit?" but they are very bad at answering the question "what polynomial best approximates f(x) over its domain?".

If you want to answer the latter question, you should use linear algebra and change of basis / orthogonal filter banks / whatever you want to call it. This is the trick used by the Fourier Series and the Discrete/Fast Fourier Transform. It works for polynomials too: google "Legendre polynomials" or "Hermite polynomials" for more information on polynomial analogs of the Fourier Series (if you haven't had linear algebra, I highly recommend Axler's book, which will teach you how to use generalized inner products to perform this trick). There's a second type of polynomial approximation called "Lagrange interpolation" which approximates a function using polynomials by evaluating it at certain points rather than by integrating it. Lagrange interpolations are only distantly related to Legendre/Hermite/etc polynomials but both strategies answer the question of "what polynomial best approximates f(x) over its domain?" much better than Taylor series.

Also, if you ever want to learn how MP3/JPG compression actually works, you'll want the DCT, which is another relative of the FFT. If you know the relevant linear algebra, the distinction is trivial (DCT has implied "mirror" symmetry at the endpoints rather than "wrap" symmetry so you don't get ringing if the color changes from one edge of a block to the other). DCT, FFT, Legendre polynomial approximation, Hermite polynomial approximation, etc all use the same simple trick (cue "one weird trick" ad). If you don't know the relevant linear algebra, it's a bunch of black magic. But the relevant math is accessible enough that I highly recommend looking into it. I've already mentioned Axler's book (Linear Algebra Done Right) -- it's accessible to anyone with a semester or two of calculus under their belt and it'll get you to the point of understanding how all of these transforms are basically just rotation matrices for high-dimensional spaces (one dimension for every point you evaluate the function at) within a few chapters.

The key is to understand the distinction between vectors/operators as understood by mathematicians and the blocks of numbers called vectors/matrices by engineers. The latter are like a shadow cast by the former. Learn about the former and you'll see that the DCT, FFT, etc are shadows of the exact same concept as the familiar triplets of real numbers, just cast at a different angle, if that makes sense.