The book has been written by David J. Griffiths, and I use the one from 2005.
Word of warning: you cannot appreciate QM without some understanding of mathematics. I don't know your level of knowledge on the matter, but chances are that some techniques are new to you.
The "DJ", as this guy is called among physics students, wrote a nice hands-on book, guiding the reader into the topic with his active and humouristic style. Lots of exercises help the student learn-by-doing.
The true power of QM lies in the fact that it works so well, mathematically and experimentally. Broader interpretations on why this is the case are subject of discussion to date. However, materials, magnetism, lasers, and so on, are enormously accurately described by quantum mechanics. That is why Griffiths starts with getting one acquainted with the application of the techniques, before discussing Schrödinger's Cat or the famous Bell Inequalities.
Griffiths starts off with laying emphasis on the importance of understanding what quantum mechanics does, before discussing what it means. I think, that is important, as the probability that two readers have the same entrance level is quite small and to be able to discuss amongst experts, the How question comes before the Why, sometimes. And those times can be really interesting indeed. That is what Quantum Mechanics is about.
Evert competent physicist, can "do" quantum mechanics, the DJ writes. And for I am trained as a physicist, it would be proper to handle quantum mechanics primarily in the way it originated.
So in my story, we start with Planck. In the book, after the preface, he starts with the Schrödinger equation. Yes, there are equations in the book. And maybe I will write about them in an appendix. Because that is the language quantum operates in, in mathematical language.
Here a little story that Griffiths does not tell you. It is the story about Max Planck. He was young and wanted to do his doctorate. So he came to his professor who strongly disencouraged him to do physics. "The world is known, just a few glitches. It's not interesting.", the professor said.
But Planck's eye did fell on a topic of black body radiators.
So first let me tell you something about black body radiators. Picture a (black) spherical mass. When it gets hot, by receiving photons and absorbing them, it starts to radiate heat waves.
Now the frequency with which the outcoming photons radiate has something peculiar: it comes in steps. From their nature however, photons are expected to come in waves, with a continuous presence. Steps are discrete. Like from one sport on a ladder, there is a fixed height to another.
Many objects can be modelled at a black body radiator, e.g. the Sun. The color spectrum (visible, infrared) of the star is a result of the black body spectrum of its surface.
A picture sometimes says more than a thousand words. Looking at a drawing or a graph can be really insightful.
The next topic of interest is the wave function. Each particle has one. It describes the probability with which you expect to find a particle at a given place at a given time.
The Schrödinger equation describes the dynamics of the wave function.