I'm a hobbyist pianist who has been playing primarily rock and blues music for about 2 years, and classical before that for a few years. I have a beginner grasp of music theory (I know my major, 3 minor, and blues scales, and work on ear training and understand intervals). I thought to branch out in my playing (especially improvisation for cocktail-style piano) and get some further understanding of music theory by reading through the Jazz Theory book, however, it's pretty tough to get through for me. Given this book is nearly straight theory and has very few exercises, should I be working through other books more my level before diving into the heavy theory?
Yes. That book, while being very good, is not paced well for beginners. It throws a ton of hard-to-learn stuff at you right away. I'd recommend supplementing it with other books that perhaps have more written out examples to work with as well. I wouldn't say it's good as an only book.
The thing about learning jazz is that I could sit down with a new player and in a half hour show them exercises that would take five years to master. But that doesn't mean that's a good idea...
Have you tried Charles Austin's An Approach to Jazz Piano? It's probably the most comprehensive jazz piano/jazz theory book out there, and it starts with "rock" type harmony and moves on from there. I found "The Jazz Piano Book" assumed a lot of cultural knowledge that I didn't necessarily have at the time, when I was starting jazz piano, as well as introducing ideas in a weird order, and at the time I just needed a really thorough explanation of all the things from first principles, and ways to put them into practice.
I've had the Jazz Piano Book for 30 years and just started looking at it very slowly and just a little bit at a time.
I think it is a great book. There is just a huge amount of material that is presented in a very compressed manner so taking it very slowly and methodically is helpful.
For instance - in one chapter on melodic minor chords he shows 5 different left hand voicings for a Dmin7b5 chord. It takes up only about 1/8 of the page. So it is easy to read it and skip over it but what one really needs to do is to learn all 5 voicings in 12 keys. So that is 60 different chord voicings each of which looks different and feels different from each other. The real goal is to be able to learn what each of these voicings looks like, feels like, and sounds like. This takes awhile. But while learning all these chord voicings one can also try and use them in songs out of a real book so one can learn what they actually sound like in context.
I found that it is definitely worth it because by doing it slowly and somewhat painstakingly, over time I began to see the mathematical (not calculus, just arithmetic!) relationships among the various chords and scales. And those relationships stay in my mind.
Levine's The Jazz Piano Book is more of a textbook than a How To, so no wonder you are finding it hard work. Given that your stated aim is to get into 'improvisation for cocktail-style piano', perhaps start by obtaining jazz piano arrangements of some cocktail-style tunes; you are bound to learn something just by playing them, and the cocktail quaffers are unlikely to care whether the solo is yours, Hal Leonard's or Mulgrew Miller's (Levine is a big fan).
Next, it's a matter of experimenting with different chord voicings and slipping in a few substitute chords to put your stamp on the tune. It's probably best, when starting out, to confine your experimentation to the solo section, rather than the head, so as not to cause too many ripples in the martinis. As Laurence Payne says,'...they still want to hear the tunes...'
The next step is pretty much mandatory: Get The Real Book Software, which contains a lifetime's worth of jazz tunes and three lifetimes' worth of 'definitive' recordings. You then listen, listen, listen, transcribe, transcribe transcribe and play, play play. Then transcribe some more. You can cut to the 'cocktail' chase and skip over the more frenetic tunes; that will leave you more time for transcription. Transcription is critical because it is your link to a century's worth of innovations by jazz giants.
Sooner or later, the listening and transcribing will put you in a position to make more sense of the theory when/if you return to Levine's book.