Should I begin to study quantum mechanics or wait until I have a stronger base on easier topics? I apologize if this is off topic since it does not relate directly to the study of physics, but I could not think of a better place to ask.
I am a high school student graduating in a few days and have recently developed an interest in quantum mechanics. I began to study physics about one and a half years ago and calculus two years ago, and I feel confident about my abilities in integral and vector calculus (at a typical college Calc I–III level), differential equations, mathematical problem solving, and some set theory. I have been introduced to quantum behavior, particle physics, and nuclear physics, but none of these were delivered with a heavy focus on mathematics. Quantum physics, however, caught my attention as a future field of study due to its bizarre nature and inquisitive mathematical models.
Without prior knowledge of more difficult topics, I am concerned that attempting to study quantum mechanics will only provide entertainment rather than benefit my college career. I have selected a few books (Griffiths, Sakurai, Feynman, etc.) of which I will choose one to read over the summer of this year and onto college if I determine to do this.
Should I follow these plans without first strengthening my knowledge in other fields of physics and mathematics, as I dive directly into the topic? Or should I wait until I am more experienced in other fields before discouraging myself with the difficulty of quantum mechanics? If so, what areas of physics or mathematics would you recommend as prerequisites to the topic?
 A: I have had to take two or three swipes at every technical topic I learned. Yes, more electromagnetism, vibrations and waves, and theoretical mechanics, along with linear algebra an Fourier analysis will make it easier to learn quantum mechanics. 
Even if you study those things first, though, you will still probably need to study quantum mechanics more than once. I'd say take a shot at quantum mechanics now, then study some more and come back to learn it more deeply. 
Check out Thomas Jordan's "Quantum mechanics in simple matrix form" for a simplified but technical introduction to quantum mechanics before trying Griffiths. 
A: Certainly with a basic knowledge of calculus you should be able to grasp some basic ideas of quantum mechanics.
The best book I've ever read on quantum mechanic is Shankar, the first chapter is absolutely solid maths, but once you get over it, it's all very straightforward.  The maths itself is very well explained, and I would imagine a first year undergrad wouldn't have much of a problem.  Buy yourself a copy of the Feynman Lectures anyway, they're invaluable for anyone interested in university level physics.  Be wary of the high-ranked books on Amazon, many of them are excellent texts, but only much use at graduate level (Sakurai, for instance).  Most people recommend reading a "soft" textbook like Griffiths before tackling the classics.
It's not entirely necessary to know other fields of physics before you dive into quantum mechanics, but as with everything, knowing the bigger picture helps a lot.  Certainly when learning quantum mechanics it is invaluable to be able to compare its predictions with those of classical physics.  You needn't know much electromagnetism besides basic electrostatics, pretty much all I've ever used as an undergrad in QM was the Coulomb force and dipoles (when dealing with spin-orbit and the Zeeman effect).
The other answer about multiple coats is pretty much spot on.  Most of the QM I've been taught started out with a statement or observation and the actual explanation came a year later.  QM can very quickly get very complicated and most of the major results are very subtle, the Exclusion principle, for instance.
However, the only way to find out would be to dive in and go for it.  If you find it's not for you then you haven't lost anything.
A: You seem to have the right mathematical background to take QM head on. But the following readings will make it more enjoyable.
The Feynman Lectures on Physics

Classical Mechanics

CM will introduce you to a lot of new ways to look at Newtonian mechanics. This is not solving pulley problems and sliding weights off wedges. No, this introduces you to linear algebra and powerful principles such as Lagrangian mechanics, calculus of variations, Hamiltonian mechanics, chaos theory etc. I found the generalization of Hamiltonian mechanics to QM very elegant.
A: All of those references will assume that you know certain things about mechanics and E&M as well as a certain amount of math.
The result is that you could bump into a line like

It clearly follow that...

and something that seems totally opaque to you.
If you are willing to risk this and able to take hours or days off while you go figure our why that should be "clear", then there is nothing stopping you from diving right it.
If you prefer things presented in a more considered and orderly manner you might want to start at the beginning.
A: When I was a high school student (and probably not as knowledgeable as you) I remember studying "A Quantum Mechanics Primer" by Gillespie. It is all about grasping what do the postulates imply, working with wave functions. Probably there are better ways of introducing the material right now, and more physically motivated, but I remember fully enjoying (and studying in every detail) that little book.
On the other hand, from a very different point of view (working with discrete spaces, e.g. state vectors, so that all the math is algebra), "The Strange World of Quantum Mechanics" by Styer is also a quite nice little book. This is the preferred type of presentation today, as many introductory books (and lectures) on Quantum Mechanics work with discrete 2-state spaces (e.g. spin or polarization).
A: Go for it. Griffiths would make fine summer-time reading.
Alternatively, you might consider making the Feynman lectures your summer reading.  They provide a great deal of insight but are better to read before (or at the same time) you take the intro courses.
A: I think that if you are to really develop a sense for physics, you must spend time cultivating a physics intuition in addition to mathematical prowess, so if you're motivated enough to take a crack at Griffiths, take time to investigate other areas, like E&M and Statistical Physics, and try to really relate the ideas you find there to things that you already understand.  
Also, spend some time with a computer language, see if you can create simulations to help your understanding.
If you have iTunes, there are a number of lecture videos at iTunesU that discuss quantum, and also the MIT Open Course Ware Site:Physics has a number of very courses with lecture notes and some with full video sets.
Good luck.
A: As others have stated or suggested, there is no right answer, although I get the notion that some feel there is a proper order to these things, and being a layman I am not qualified to judge what is proper when it comes to physics education.  However, I will offer you this nugget:


*

*Understand the difference between an
equation and a solution.


Some may laugh at this statement, or even shake their head in empathetic embarrassment, but I will tell you that one of the hardest concepts people have grasping (even extraordinarily intelligent people) when starting in QM is what is meant when one says Schrodinger wave equation and what one means when they say wave function.
$$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) \
 = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t)$$
$$E\Psi(\mathbf{r}) \
 = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r}) + V(\mathbf{r})\Psi(\mathbf{r})$$
Above are time dependent and time independent Schrodinger equations.  Most people would not be able to distinguish which component of the above expressions is the wave function.
In the first case its (the time dependent case):
$$\Psi(\mathbf{r},\,t)$$
In the second its (the time independent case):
$$\Psi(\mathbf{r})$$
Without this distinction it is very hard to understand statements like:
Solutions to the wave equation are wave functions which represent the probability amplitude of a particle being found at position r and time t.
or;
The time independent equation describes the standing wave solutions, or the energy eigenstates (states with definite energy), of the time dependent equation.
The simplest way I have in conveying understanding, without being overly pedantic, is to explain to a person that functions are simply all the possible lines one can draw on a paper that do not cross themselves (like in a loop) and do not make abrupt changes, jumps or skips.  The wave functions are those sets of lines that also satisfy the constraints place on them by the wave equation's operators; e.g. the linear differential operators like:
$$i\hbar\frac{\partial}{\partial t}$$ and, $$-\frac{\hbar^2}{2m}\nabla^2\ + V(\mathbf{r})$$
I think understanding these distinctions transcend most physical subjects, even if the specific equation is not related to QM.  So I think it is something that one should feel comfortable understanding, and if one uses QM as the context I think it is o.k.
I would like to add, I am not trying to pick on the lack of understanding of these things by my fellow laymen, I just think that it is something that represents the first initial barrier for most people, and thus is a source of discouragement.  So if anything in the above can help aid people, I think it is a good thing to provide as input.
A: My suggestion to the OP would be to read Dirac 'Principles of Quantum Mechanics'. It is very clear, and teaches you liner algebra in a physical approach. Also, I think it best conveys the thrills and motivation of doing Theoretical Physics. You could start reading it right away, without trying to read any other maths without motivation. 
