I would like to to beyond my current understanding, which is basically “they’re effectively one-way functions”, and be able to participate in discussions of articles such as this one.
For the design and internals of hash functions? The finalists for the SHA3 competition have extensive design documentation. There's an archive at https://web.archive.org/web/20170829225940/http://csrc.nist....
Cryptographic hash functions are designed to resist existing attacks, so you'll want an understanding of differential & linear cryptanalysis, as well as a variety of algebraic attacks. I don't know of a good textbook on the subject, so you might find yourself searching keywords on https://eprint.iacr.org/