I can only agree that you first should study non-relativistic quantum mechanics, before you go over to relativistic quantum field theory. You need to learn the concepts of quantum mechanics. The math is not so difficult. If you mastered classical electromagnetism, you are well prepared for the beginning of a quantum-mechanics course, using the wave-mechanics approach, which I think still is the best way to introduce quantum mechanics.
The problem with learning (and also teaching!) quantum theory is that it requires to rethink the whole picture about nature completely. Against this the already unusual features about relativistic space-time, which is more about physics in general, including classical (i.e., non-quantum) physics, is easy.
I don't know the book by Griffiths, but what I know from his writings in Am. J. Physics (if this is the same Griffiths ;-)), I guess he must write good textbooks.
My favorite for beginners is J. J. Sakurai's Modern Quantum mechanics, followed by S. Weinberg's brand-new book. Then one should also have a look at either Ballentine's book or his Review of Modern Physics article on the Minimal Statistical Representation.
For those, interested in the foundational problems of quantum mechanics, I recommend the very "down-to-earth" and carefully written book by A. Peres, Quantum Theory: Concepts and Methods.
The latter is not necessary, however, for quantum field theory. There are many good books on the market. Zee's book for sure is not among them, I'm sorry to say. On an introductory level, I'd rather recommend Srednicky, Ryder, or Peskin&Schroeder. The best books on the subject are Weinbergs three volumes "The quantum theory of fields".