I have spent a TON of time in the math library at Rutgers just thumbing through all the books, and I never really found a book about the "math mindset" that was half decent. You could do a lot worse than to read stack exchange every day and become a regular on this forum; everyone here is really helpful and you shouldn't be afraid of posting a "silly" question out of fear of being embarrassed; the only thing that upsets people here is when someone doesn't put a sufficient amount of thought/effort into the problem before asking, or else when the question being asked is clearly off topic.
I think that talking with mathematicians about specific problems you are working on will give you a far deeper insight into how mathematicians think than reading some low level book for a popular audience will.
The most important thing in my opinion is not the book you finally choose, but rather the attitude that you should always try to solve EVERY SINGLE problem in whatever math book you read. You should spend more time solving the problems than reading the sections.
I recommend you pick up "A survey of modern algebra" by Birkhoff and Mclane. The difficulty level is pretty low, and the problems are numerous and instructive. One reason I so highly recommend this particular book for you is that "modern algebra" is not only fundamentally important for contemporary mathematics, it also is not taught in high schools generally, not to mention that many of the results are beautiful. Another great thing about this specific book is that the author's are famous mathematicians, and you can figure out how they think about math just by seeing the organization of how the material is presented and how the results are derived. The first chapter in this book deals with number theory, but actually contains a very nice introduction to writing logical and rigorous proofs.
I think you should just read through books on specific subjects, and pick the subject you will study based on your current interests and level of knowledge. Some subjects to start with are:
set theory, metric spaces, complex analysis, group theory, number theory
One thing you might want to try is just to go ahead and check out some books by the great masters. For example, there is a great set theory book by Frankel (who has the axioms of set theory named after him), and a great book on number theory by Sierpinski; and I recommend both of these not only because they are really well written and basically at the high school level, but because this will give you an insight into how the great masters thought about math. Finally, if you are feeling confident go ahead and check out Disquisitions Arithmetic by Gauss, just to try your hand at it.
Let me also mention a compilation of writings for a popular audience by Henri Poincare called "The Value of Science". A large portion of this book deals with physics, but there are two particular passages you should pay attention to for understanding the "math mindset": the first is the discussion of induction, and the second is a fairly famous section where Poincare discusses the internal psychological process he experienced when solving a highly technical problem.
Best of Luck!