I understand that given that expanding a function onto polynomials is a valid thing, the equation for Taylor series follows, but why is expanding a function onto polynomials a valid thing to do? Why should we assume that something can be written as $a + a_1(x)^2+...$. It doesn't seem inherently obvious to me.
I think you need to take a look at Stone–Weierstrass theorem. See http://en.wikipedia.org/wiki/Weierstrass_approximation_theorem and any standard analysis book (Rudin will be good enough)