On July 6, I gave a summary of what I thought made a good introduction to classical physics. This week, I'm writing on those topics which caused a revolution in the science, and distinguish modern physics from classical: relativity and quantum mechanics. A good resource is the website of John Baez (click on the "Fun Stuff" link at the top). Another nice resource is av8n.com, the website of John Denker. In particular, he has a wonderful article on whether gravity is a pseudoforce or not. He also has a (much longer) essay which clarifies many points of confusion with thermodynamics -- but which I must emphasize is not an introduction to thermodynamics; his points wouldn't make sense without a preexisting context of thermodynamics and statistical mechanics. A word of warning, however: he has no reserve about coining words and phrases when the commonly-used terms can lead to confusion. In particular, he calls exact differentials as grady differentials, and he emphasizes the difference between traditional thermodynamics courses with what is seen in practice by calling the former cramped thermodynamics; the difference is explored throughout the entire article. I want to emphasize section 6.10, which describes the use of one-forms for integration, rather than the usual dx notation; an understanding of this is important for those who wish to go on to general relativity, and who therefore need to understand general relativity.
A person can start from either special relativity or quantum mechanics; but since the mathematical requirements for special relativity are so much simpler, I will start my overview there. According to various sources I've used for learning special relativity, the key to understanding the subject is a mastery of Minkowski space-time diagrams. A very nice book for this is John Kogut's Introduction to Relativity, which uses nothing beyond trigonometry for developing the basics of relativistic motion and demonstrating the use of such diagrams. (However, derivatives are used for relativistic kinetics, to derive the basic formulas for mass and momentum.) Starting from the two assumptions that the laws of physics are the same in all inertial frames, and that there is a fixed speed no object can exceed, he proceeds to demonstrate in clear, logical steps how the results of special relativity follow, from time dilation to E = mc^2. The book does stop short of electromagnetism, and a demonstration of how the unification of electricity and magnetism results from special relativity; I will look around for a good resource on this topic. The book also gives a small preview of general relativity in its last two chapters, which is nice.
As for quantum mechanics, there are three mathematical approaches to the subject. The first is to represent a system by a function (called the system's wave function) which contains all obtainable information about the system; and anything that can happen to the system is represented by an operator. The second approach is to use harmonic analysis to represent these wave functions as (infinite) vectors, and the operators as matrices. These two approaches are used in Robert Dicke's Introduction to Quantum Mechanics, a book I recommend in part because it gives nice refreshers on topics in math and physics when necessary; as well, Dicke ends his book with a chapter on quantum statistical mechanics -- simply enough, statistical mechanics where the particles act quantum-mechanically. (I must note that it gives little room to mentioning the Einstein-Podolsky-Rosen paradox, though I'm undecided whether this is good or bad for an introductory text.) The third mathematical approach, the path integral approach, considers the various possible paths a particle might take as "points" to be integrated over; while mathematically more advanced (requiring functional analysis), this approach is more general.
Quantum field theory is another subject I wish I would have learned about earlier than I did, and I have the people over at Panda's Thumb to thank for first making me aware of it. Quantum field theory can be considered as a synthesis of quantum physics and special relativity (so that when physicists talk about difficulties between quantum physics and relativity, it is general relativity they mean). Where Schrödinger's equation results from the Newtonian expression for kinetic energy in terms of momentum, the relevant equations in quantum field theory are derived from the relativistic energy-momentum equation. Beyond this, I'm still in the process of trying to understand the subject; though I do know that while the solutions to the Schrödinger equations are functions, the solutions to the equations in quantum field theory are considered operators. The books I'm using for the subject are both called Quantum Field Theory, one by Franz Mandl and Graham Shaw, and the other by Claude Itzykson and Jean-Bernard Zuber. The book by Mandl and Shaw is simpler to understand, but does not go over path integrals. The book by Itzykson and Zuber does cover path integrals, but I find its development more difficult to follow; also, not counting appendices, it contains 690 pages to Mandl and Shaw's 329. Another difference is that Mandl and Shaw's book has problems at the end of its chapters, whereas Itzykson and Zuber place unlabelled exercises throughout the text, mostly to prevent the repetitiveness of very similar cases. Mathematically, at least with Mandl and Shaw, the requirements don't seem to be much beyond that of Dicke's book; it just requires a solid grounding in quantum mechanics and special relativity (including relativistic electromagnetics).
I come finally to general relativity, which I have left for last because by my judgement, it requires the most higher-level mathematics. Not only does it involve curved spacetime, which necessitates knowledge of differential manifolds; but the special spacetime structure of special relativity is incorporated by Lorentzian metrics, which I'm not very comfortable with. A good introductory textbook on the subject should include a chapter on Lorentzian manifolds. Also, general relativity does not require familiarity with quantum mechanics, so if you want, feel free to start on it once you are secure in your knowledge of special relativity.
That is it for my overview to physics. I was wanting to find some good resources on Lagrangian mechanics and on continuum mechanics, but have not had much success. I could probably write one myself on the former, but I have not had much success on resources on the applicability and limitations of the continuum approximation. I also plan to look up information on materials science. More immediately, I plan to make my next post on biology, and then maybe a post on how science is done.