I've always found papers that collect many of the different resources, information, and references into one coherent publication to be extremely useful. This is one such paper.
P.J. Mohr and D.B. Newell, "The physics of fundamental constants", Am. J. Phys. v.78, p.338 (2010).
Abstract: This Resource Letter provides a guide to the literature on the physics of fundamental constants and their values as determined within the International System of Units (SI). Journal articles, books, and websites that provide relevant information are surveyed. Literature on redefining the SI in terms of exact values of fundamental constants is also included.
The authors included discussions on the various major fundamental constants, how each one was determined, and a list of references! The latter is the one I appreciate the most, because it tells you how these things came about in greater detail. If you don't have the patience to follow the CODATA report, this is the next best thing.