I'm a largely self-taught musician. I did take guitar lessons (from a guy who could tell what chord you were playing by ear, but had no idea what notes went into them - no theory background whatsoever), and composition/ear training lessons from a classically trained composer who got his master's at Julliard. Still, a large part of my musical education was self taught. I read a lot of books, mostly old old dusty tomes that I was lucky enough to find in reprinted editions, and rarely if ever stuff that would be considered a textbook for a classroom.
My composition teacher introduced me to Hindemith, but in his Elementary Training for Musicians and Traditional Harmony books - both of which I diligently did the exercises in every week.
This is by far my favorite book on the subject of composition. While there's plenty that I've learned that isn't in this book, there's so much material in here that really strikes at the heart of harmony, that it's hard not to love it... or leave it, as the case may be - a few reviewers here seem to regard it as more of a cultural oddity than a book that shines a light on the dark places. Perhaps these reviewers have had the benefit of a traditional musical education & have spent years with far thicker tomes, such as Piston's perennial classroom favorites.
He begins (as did many theory books up until Hindemith's era) with an overview of the overtone series & the creation of the major scale. He goes one step beyond that, and through a series of convolutions, constructs a 12 tone scale that he derives from the overtone series (Rameau would be proud). I see the merit in this, but it seems largely academic to me. Either you'll use equal temperament in atonal composition, or you'll compose in a largely diatonic manner. Still, I applaud his effort to marry the old tonal tradition with the emerging atonal compositions that were becoming more prominent in his time. Few of us today are concerned with the precise place we need to put our fingers on the fretboard to get a properly intonated minor second, but in his day when composers were transitioning away from purely tonal music, but were not embracing equal temperament, this would have been an issue.
It's worth noting that Hindemthi hated equal temperament & is glad that he doesn't believe it will take over music any time soon... published in 1942, two years after Les Paul built his first electric guitar, and less than a decade away from Leo Fender transforming the world with inexpensive, mass produced electric guitars, transforming the popular music world into one of equal temperament seemingly overnight. I find this both sad (because I suffered for many years as a musician because of the equal temperament music I grew up on was not a harmonically pure enough one for me to develop a good ear early on), and ironic, since he was on the cusp of a revolution & didn't know it.
When he's done constructing the scale, he uses combination tones as a justification for the way chords are constructed - that is, if you have 3 notes, how can you definitively say one is the root? Even when in inversion, why is that note the root? To those of you who know nothing of the overtone series and think of chords as "stacked thirds" - I ask you - why is the root note the root? Why can you have another note in the bass, yet it's still not the root? How can you add a 7th and a 9th so that your chord has 5 notes (5 out of a possible 7 notes in the major scale), and then invert it, and still know which note is the root? Hindemith explains.
From there, he places the intervals on a continuum from more harmonic to more melodic. I don't think his groupings are well accepted dogma (nor any of the things I'm talking about), but it makes good, strong, logical sense. If you play a succession of tones, why does playing 3rds and 5ths sound like you're arpeggiating a chord, but playing seconds sounds like melody? Again, Hindemith explains.
Building on the above, he then breaks chords into different groupings. This may be less controversial, but his method is logical. These groupings make up the entirety of the two page appendix (unless you count his section on analysis of musical samples as part of the appendix too), so you can be sure that Hindemith believed these groupings were important for you to have on reference (though it's fairly easy to remember the groupings - simple triads, 7th & 9th chorsd, chords without a tritone, etc.).
This takes up roughly the first half of the book, and lays the foundation for the second half - Hindemith's rules of Harmony. This bit is somewhat more detailed & technical, and I found it to be less "full of insights" than the first half, and more "full of rules" - which is what a great many harmony books are, so this section was a bit less valuable to me.... I say this after having read the book a year or so ago, so mostly I'm saying this because I remember fewer of the thing he said in this section, I may have been rapt in attention during this section a year ago, I just don't remember it so well now.
There's also a brief section on Melody. Very few music theory books tackle melody, and Hindemith does, which is very laudable. I don't know that he adds much of significance here, but it's certainly worth going through this section, since so few treatments of melody exist.
Finally, we have the aforementioned analysis of musical works, and the chart of various chord types.
I found it to be quite an enjoyable read. Very concise and with little fluff, and for the most part, lite on "rule" and heavy on "insights into why thing work" - which is the sort of stuff I live for.
If you want more "rules on classical harmony" Tchaikovsky's book is slender and hard to beat. If you're really just looking for "music theory for dummies" Edly's Music Theory for Busy People (available on the publisher's website) is very simple & concise, though the first few pages may be difficult for an absolute beginner. Writing Music for Hit Songs is also a great introduction to music theory (and from a pop music, not classical) standpoint, and I would recommend it to absolute beginners.