What's the standard "roadmap" to learning quantum physics? I'm really interested in quantum physics and would like to learn more. However, I don't know where to start and in what order I should learn things. So, ideally I'm looking for some sort of roadmap of what to learn. What physics topics do I need to know to start learning about quantum mechanics? (In addition to the mathematical topics mentioned at What is the math knowledge necessary for starting Quantum Mechanics?)
My current knowledge is mostly popular science stuff, like tv shows on Discovery Science and National Geographic Channel. So I have a basic understanding of some basic principals. There's also a recent lecture from Brian Cox that I have watched which gave a bit more in-depth information.
 A: Most of the other answers are discussing the mathematical "prerequisites," but let's try to describe a standard roadmap. Most (American) universities (that I have experience with!) have a sequence for physics and brave engineering undergraduates that begins with three courses in sequence, the only math you need generally being a firm grounding in high school calculus:


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*Newtonian mechanics

*Electricity and magnetism (not quite at the level of Maxwell's equations)

*Modern Physics (Includes special relativity and the basics of QM)


At that level, basically after a year of very relaxed training, you'll know enough about quantum mechanics to be dismissive of QM claims that pop up on the Discovery channel! If you want to be an expert, then certainly much more work is required, but I'm a firm believer that you can learn a lot about a topic without knowing everything...
This leads nicely into mathematical "prerequisites," and I use scare quotes because I think it is easy for a beginner to get overwhelmed by the idea that they basically need a degree in mathematics before they can understand physics. Every single one of the topics that Jose Moreira mentioned in his answer will help you understand QM more deeply (and be remiss not to recommend Strang's book for linear algebra!) but you'll learn the bits you need as you go. One of the best quantum mechanics books for undergrads -- totally accessible for someone who has completed the three courses above -- with self-contained introductions to the mathematical tools and a refreshingly "shut up and calculate" approach is D.J.Griffiths' Introduction to Quantum Mechanics.
A: I would suggest that you don't do any preliminary reading, and just learn QM directly. There is not much to it, the requisite background is very primitive linear algebra, and Dirac's book "The Principles of Quantum Mechanics" and Feynman's "Lectures on Physics Vol III" can be read with Wikipedia help without any prerequisites.
The classical mechanics you need to know is not very sophisticated either--- you just need to know Newton's laws, and how they come from a Lagrangian or Hamiltonian, which is covered in standard sources. You don't need so much deep stuff, although knowing Poisson brackets is handy for seeing the vestigial quantumness in the classical mechanics structure.
I would suggest reading the following Wikipedia pages for a historical perspective, which helps a lot with historical literature:


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*Old quantum theory

*Adiabatic invariant

*Correspondence principle

*Matrix mechanics
This is wrongly left out of most books, and this is a shame. There is no unified presentation of the historical material except on Wikipedia, and this is why these pages are up there. Once you get the historical stuff (it's not a lot), Dirac gives a conceptually self-contained introduction to the mathematics, the notation, and the physics, while Feynman is path-integral friendly, so you can go on to read Feynman and Hibbs, or Mandelstam and Yourgrau without any delay.
It is usually a waste of time to try to go through prerequisites, as these are usually boring and most of the material doesn't end up getting used. For QM, you need to come in knowing what a matrix is, and what an eigenvalue is, which is probably best learned from Dirac.
A: I don't think there is a standard roadmap because it depends very much on what you're trying to achieve. I first learned QM when doing a Chemistry degree, and that was heavily focussed on the Schrodinger equation and time-independent solutions. My friends doing Physics took a more abstract and mathematical approach.
From your question, I'd guess you're looking for something like a traditional popular science book but more rigorous. The trouble is that QM doesn't lend itself to this approach. It isn't like general relativity, where you can imagine rubber sheets and get a reasonable idea of what is going on. There isn't any simple mental image of QM you can work with. The only good way to learn it is to get stuck into mathematics.
You might want to have a look at Susskind's lectures for non-physicists. See http://www.youtube.com/watch?v=2h1E3YJMKfA for the first lecture and search YouTube for the others. I thought they were rather slow-moving and only had the patience to watch a couple of lectures, but they seemed to me to give a reasonably simple but rigorous introduction to QM.
A: Like Rennie said there is no standard roadmap for learning quantum mechanics. Even though there is no standard way of learning QM I'll show you one of the possible ways of doing so. I'll assume you have the basic high school math tools:
You will need a basic calculus course, one good book for that is Calculus, Stewart a general and simple but a good approach. Take a look at the MIT calculus courses (I and multivariable calculus); 
Then you will need at least one general linear algebra book: Linear algebra from Hoffman would be my choice but Kunze is also very good.
Then you should have a notion on probabilities and statistics. For that, I would suggest  John E.Freund Statistics the first course as an introduction and then you should pick up one more heavy probabilities and statistics book you like. the one I mentioned won't be enough in my opinion.
You will need to know a book about differential equations "elements of ordinary differential equations and boundary value problems" will do just fine even though it is a bit introductory a more complex book after that one would be good.
Then to finish your mathematical preparation you will need to know Fourier analysis for that I suggest for a basic approach a book not exactly about Fourier analysis but a book that has a very good approach (in my opinion) Signals and Systems, Oppenheimer. 
If you study and Understand all the topics above you will be able to learn QM with no problem. A good with a somehow historical introduction to QM is Modern Physics from Krane but this one is just a transposition from classical to QM not really a QM book. As a QM book quantum mechanics, Claude Cohen-Tannoudji is a good book.
There are two things I must say:
1st a knowledge of the mathematical tools isn't enough you will need to know Newtonian Mechanics and Electromagnetism along with an introduction to relativity AT LEAST.
2nd if you do want to learn QM be aware that it will shape the way you see the world for it will show you how little we know about it. It will show you how complex the laws governing our universe are. 
Good luck.
A: For  theoretical  purposes  I would also  recommend  you  the  following :


*

*Principles  of  Quantum  Mechanics  by  P.M. Dirac.

*Mathematical  Foundations  of  Quantum  Mechanics  by  J. Von  Neumman.

*An  Introduction  to  Quantum  Theory  by  Keith  Hannabus.
Now  because  of  your  interest ( according  to  your  profile )  on  computer  science,  you  may  want  to  read  something  about  quantum  computing. For  example  "An  Introduction  to  quantum  computing"  by Phillip Kaye, Raymond Laflamme  and  Michele Mosca.Or  the  following  arxiv.org  material :
http://arxiv.org/abs/quant-ph/9809016
http://arxiv.org/abs/0708.0261 
Good  luck, I  am  also  trying  to  learn  Quantum Physics ! :)
A: I think the usual way as they take it at a university is pretty much the time efficient way to go. The math and$\ $  *at least*$\ $ Classical Mechanics will be necessary.
Nobel Laureate Gerard 't Hooft at one put together a selection of introductions in a roadmap style 
$$\text{click}$$
The navigation system on that site doesn't work for me anymore. But the important part, the links on the right hand side, work.
A: I'm sorry to tell you that anything you learn on TV about quantum mechanics is useless. Not many people in the world understand quantum mechanics and many of those who claim to, are either naive or liers. I don't understand quantum mechanics, but I could calculate your things, even explain a model of the hydrogen atom. Quantum mechanics is not "just" understanding, it's about BREAKING all your concepts, it will destroy every notion you have up to this moment. With this said, I recommend you this college-like path:


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*Algebra and analysis: vectorial spaces, integrals, differential equations are BASIC, so go deep into them, specially vectorial spaces so you can "upgrade" to tensors from there.

*While learning the basics of algebra and analysis you should learn about Newtonian mechanics and maxwell electromagnetics (in that order).

*Once you have done it, learn everything about Newtonian mechanics again using the Lagrange and hamilton formulation. At that moment you will be able to mix maxwell electromagnetics and Newtonian physics.

*If you really want to go deep you should make some digging in the special relativity, but that is somehow optional... (not if you want real understanding).

*Now start with old quantum mechanics books, not very similar to modern physics, but absolutely necessary if you want to understand WHY did we go the way we went.

*At this point you actually know something about QM and you are ready to learn the Dirac principles and finally destroy every concept about the reality you had before starting this path.
This could be a 2-4 year project, but it is possible. I would recommend you go to lessons (in my country you can go without paying, but they won't give you the certificate). Good luck.
Note: after ALL THAT, you will get a lot of profit by reading the EPR paradox, Bohr & Heisenberg opinion about quantum mechanics, and of course the Bell article about entangled particles. Don't get mixed up by von Neumann's principles, in my humble opinion, I think he was way too mathematician to understand the deep of Bohr and Heisenberg thinking. And by the way, if you ever don't understand something, you are doing it right.
A: If you want to do research in theoretical physics, you have to be very strong in math (abstract math).  Otherwise, if your goal is to understand what's going on in quantum physics, field theory etc, then you will not need any math beyond Boas' mathematical methods.
A: It is possible to learn Quantum Mechanics before taking Classical Mechanics.  That is what I did.  But you do have to know what a vector space, basis, change of basis, linear transformation, and matrix are.  Then read Dirac.  You can learn the Hamiltonian point of view from Dirac.  So I would say that the prerequisites are: Irving Adler, The New Mathematics, and Isaac Asimov, The Realm of Algebra.
