I can highly suggest "What Is Mathematics?" by Courant, Robbins and Stewart.
You don't have to read it cover to cover, but this is the book that will be very helpful in shaping the right perspective.
As a side note, a friend of mine gave it to me in high school. It was probably the book that influenced me the most mathematically. I ended up getting a PhD in math last year.
I like this book too, but I'd say for a typical reader it's more like something you'd want after getting to the point where you're caught up and ready for university mathematics. It's a fairly demanding book. I'd recommend checking reviews on Amazon or something to get more perspectives before diving in.
It is a demanding book -- I never got to really finish reading it cover-to-cover -- but it's very helpful nonetheless even if one reads the parts they find easy.
For example, the first dozen pages on Calculus have imparted a better understanding of the subject than all the indecipherable (if only due to sheer volume) tomes of Stewart. It was an invaluable boost.
You don't have to read it cover to cover, but this is the book that will be very helpful in shaping the right perspective.
As a side note, a friend of mine gave it to me in high school. It was probably the book that influenced me the most mathematically. I ended up getting a PhD in math last year.