Where can I find a comprehensive treatment of this important result at the level of a very advanced undergraduate/beginning graduate student? What works develop the relevant material in a cohesive and readable way? Assume only a familiarity with basic homological algebra and ring theory on the part of the reader in assessing this question, please. Thank you!
2 Answers
The place I learned it from is Chapter 19 of Eisenbud's Commutative Algebra. Most of the proofs in that section do not use material from previous chapters if I recall correctly. The route (which I think is what you are looking for) is to construct the Koszul complex of the residue field of a regular (graded) local ring and also prove the symmetry of the Tor functor, and then use these two facts to get finite global dimension which implies Hilbert's syzygy theorem.

1$\begingroup$ You need to construct the Koszul resolution $K\to k[X]$ of the polynomial ring $k[X]$ over itself as a bimodule over itself and show that it is in fact a resolution. Using it, you can construct a projective resolution of the same length for every $k[X]$module $M$ simply applying the functor $(\mathord)\otimes M$, to get $K\otimes M\to k[X]\otimes M=M$. The "theory" you need is that the complex $K\otimes M$ computes $\mathord{Tor}^{k[X]}(k[X],M)$ (and this is just the definition of $\mathord{Tor}$, and enough theory to prove that $K$ is exact (for example, the Künneth formula over a field) $\endgroup$ Jun 2, 2010 at 13:35
HiltonStammbach's book on homological algebra is nice and includes that result, if I recall correctly
I am partial to CartanEilenberg's book, though!