You can also read the Wikipedia page on old quantum theory for a sketchy summary, then look at the page on matrix mechanics. This explains the intuition Heisenberg had about matrix elements, something which is not in Dirac's book or anywhere else. Heisenberg's reasoning is also found to certain extent in the first chapters of this book:
If you are not willing to learn the linear algebra upon which the entire theory of quantum mechanics is based, then you really aren't going to have much luck finding the kind of textbook you seek. It sounds to me like what you want is a textbook that introduces you to what is called \"modern physics\" instead. Most \"modern physics\" texts cover quantum mechanics concepts while remaining mostly in algebra land. Most of the textbooks recommended in the answers posted before mine are chock full of calculus.
I used this book the last time I taught quantum mechanics, and the students really liked it a lot. You can teach yourself \"real\" quantum mechanics from this book using the Dirac bra-ket notation used in real physics research and in quantum information theory.
E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, J. Spencer, Interference with correlated photons: Five quantum mechanics experiments for undergraduates, American Journal of Physics 73 (2005) 127, 153554b96e