Quite simply, these are the books I use the most with my students. No book is 100% perfect and it's often frustrating when a book has errors but these books are generally good and I come back to them time and time again.
The Jazz Piano Book - Mark Levine
An amazing introduction to jazz piano. The first few chapters are absolute gold and after you take in the theory, he soon jumps into 3 note voicings. This is a good way to start changing your voicings from dull triads to more jazzy sounding harmonies. I tend to work up to the middle then switch it up as he veers off into specific methods used by certain jazz pianists.
If you're interested in jazz then this is a solid investment:
For technical practice then this is probably the Rolls Royce of methods. It's not very fresh or original but it gets the job done. My advice is stick to one exercise and play it slowly, accurately and definitely with a metronome.
You can find all the exercises online here free:
Piano Tutor - Jeff Hammer
If you want to start from absolute zero then this is a great book. It moves a lot quicker than most methods so I recommend it for teenagers and adults rather than small children. He throws all the info/theory at you in the first 10 pages (which is a bit of an overload) and then jumps into proper notation. It takes you through treble and bass clef reading with songs from Beethoven, Abba, Coldplay, Bach, The Beatles and Johnny Cash. It mixes up styles and these are songs that you will definitely know. It's fun, a method that I endorse and it can get you from zero to grade 1 in a few months if you put the practice in!
I also recommend the rest of the really easy piano series. Nicely notated and fun to play
Dr. John Teaches New Orleans Piano
Many of you will have heard of the great Dr. John. For New Orleans Boogie and Blues piano then you can't go wrong with this book (there are actually 3 volumes).
I don't usually dig into transcription books but the CD and the notation can help to explore the fundamentals of blues and the subtle changes of style as explained by the master.
I would worry less about reading the actual transcriptions (although that can be fun) and try out some of the accompanying exercises and explanations. If you want it to sound organic then it has to come from you and you will need to make it up a bit!
ABRSM Jazz Piano Grade 1
Some people like to take on grades to focus their practice or to gain praise for the work they have put in. I'm all for it, provided that the drive is from the student themselves rather than pushy parents. This graded series 1-5 has been around for 25 years or so but I still enjoy the approach and I love that the chords and the jazz terminology is not dumbed down. It's a great insight into the world of jazz and even pop music and it can help with adding some context to your improvisation study.