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I will post, in this thread, a quick outline of the resources needed to self-study mathematics, statistics, and physics. (Edit: And basic philosophy.) EDIT If you feel that you can deal with immediate rigor, and you want more coverage than a speedy introduction to everything, skip Gerard 't Hooft's resource and MIT OCW and go straight to the textbooks listed below. SUPEREDIT I forget to mention programming. That's a super-important skill. Unfortunately I don't know anything about it, so you'll have to find help elsewhere. C++ and FORTRAN are good familiarities.

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The first is to be familiar with < >. Self-study will never work without feedback. Now, after that, there are two primary approaches to actually studying. The easiest and most recommended by far would be through MIT's open courseware movement. < > This should be familiar to most of you. To gain a strong undergraduate background in physics, you must complete the majority of the physics sequence, take at least three intermediate (real analysis, complex analysis, and algebra are my recommendations) math courses, and pick up one intermediate level statistics course (go with probability theory or stochastics). The second approach is harder but faster. < > is a website is a website made by Nobel prize winning physicist Gerard 't Hooft. It is the fastest sequence to learn physics. It will get you through the minimum necessary mathematics and statistics. It's also extremely difficult. You will need the help of Stack Exchange. The above will get you through undergraduate level classes through self-study. This much is straightforward. It gets difficult after this.
For graduate level courses, you will need to enroll in a university. Study all you want, you will not understand the important issues of science without first being exposed to research, theoretical or experimental. Enroll in a university for graduate school if you're capable (already have a BSc in anything and have taken the GRE's in math/physics). If not, undergraduate will do fine. Credit examinations will let you speed through the material you're already familiar with... this, of course, being a very romanticized vision of what's likely to happen (which is that you, being a human being, will have a limited memory of what you self-studied). Don't be afraid to retake courses. What actually happens once you get into graduate school is beyond me. There are plenty of physicists and mathematicians who are better able to help. Following this study plan, however, you should get through undergraduate's with flying colors. Self-studying math, physics, and statistics is fine, but not sufficient. Here's your reading list. Two necessary books: "Mathematics: Its Content, Methods and Meaning" by A.D. Aleksandrov, et al; and "All of Statistics", by Larry Wasserman. They will review the important things you've self-studied and will remain a strong reference well into graduate school. I recommend "The Princeton Companion to Mathematics" by Timothy Gowers, et al, if you want a comprehensive resource. It's not necessary, though; you have WolframAlpha, MathWorld, and Wikipedia. "Mathematics: Its Content, Methods and Meaning" reviews basic arithmetic and geometry; then covers analysis, analytic geometry, algebra, ordinary and partial differential equations, manifolds, calculus, complex functions, number theory, probability, approximation, computation, set theory, linear algebra, non-Euclidian geometry, topology, functional analysis, and algebraic systems and groups, "All of Statistics" reviews probability, randomness, and expectation; then it covers models and inference, CDF functionals, bootstrap techniques, parametric inference, hypothesis testing, Bayesian inference, statistical decision theory, linear and logistic regression, multivariate models, independence inference, causal inference, conditional independence, undirected graphs, log-linear models, non-parametric curve equations, orthogonal function smoothing, classification, stochastics, and simulation. Both books are extremely important foundations. Then pick up "The Logic of Scientific Discovery" by Karl Popper. It reviews fundamental problems and the scientific method; then it covers theory, falsifiability, empiricism, testability, simplicity, probability, observation, and corroboration. It's a very important book on the philosophy of science--but, I suppose, it's not necessary reading. I'll list other books that treat the subject with more rigor in the next post.

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Roger Penrose is a popular introduction into modern physics. His book, "The Road to Reality," is a solid reference. He reviews math first: basic arithmetic and geometry, jumps immediately into complex numbers; then real number calculus, complex number calculus, Riemann surfaces and mapping, Fourier hyperfunctions, surfaces, hypercomplex numbers, manifolds, symmetry, gauge and fibre bundle theory, and issues concerning infinity. After that he proceeds to the actual physics, starting with spacetime, then onto Minkowskian space, classical fields, Lagrangians and Hamiltonians, quantum algebra and geometry, entanglement, electrons and antiparticles, particle physics, quantum field theory, and thermodynamics. He finishes with speculation: the early universe, the measurement paradox, gravity-quantum theories, string theory, loop variables, twistor theory, and popular science. Warning, though--while Penrose's introduction is solid, take his words with a grain of salt. He is a well-established relativitist but a contrarian with particle physics. He also leans very heavily towards Platonian ideals. Alright, so now you've read the four important introductions. "Mathematics: Its Content, Methods and Meaning," "All of Statistics," "The Logic of Scientific Discovery," and "The Road to Reality." What now? Do you still want to do physics? I'd like to note, though, that you can do fine without reading all of these. That said, most of my professors have recommended similar sequences (with or without Penrose's book, but I think it's the best introduction. Not that I'm an expert!). Now, for heavy physics: "The Feynman Lectures on Physics," by guess who. There are three volumes. Volume one is "Mainly mechanics, radiation, and heat,"; volume two is "Mainly electromagnetism and matter,"; volume three is "Quantum mechanics." "Mechanics, Third Edition," by L.D. Landau, et al. "Electricity and Magnetism," Edward M. Purcell. "The Principles of Quantum Mechanics," by P.A.M. Dirac himself. "Mathematical Foundations of Elasticity," by Jerrold E. Marsden. "The Old Man and the Sea," by Ernest Hemingway (hehe... but yes, I'm serious). "A Modern Course in Statistical Physics," by L.E. Reichl. "The Large Scale Structure of Space-Time," by Stephen Hawking, et al. "Introduction to Elementary Particle Physics," by David Griffiths. After these books, I'm not sure. You'd be better off asking more experienced students. :P But these provide a solid background in physics. If you're up to the task--and Stack Exchange is around to help you--these are the books I'd recommend, in the order above.
Also, ignore a part of what I said above. You can skip Gerard 't Hooft and/or MIT open courseware if you wish and go straight to the textbooks. That's what I did. It's more rigorous, and more fun.
Of course, I also had the advantage of being in college, so I had professors helping me! If you really need the help, go for MIT OCW and don't skip to the textbooks.
And this concludes what I think is the most efficient undergraduate self-study guide, to complement an undergraduate education, or to preface a graduate one.
Oh, right, for philosophy buffs out there: "A Treatise on Human Nature," by David Hume. "A Critique of Pure Reason," by Immanuel Kant. "Principia Mathematica," by Bertrand Russell. "History of Western Philosophy," by Bertrand Russell. And a final, somewhat controversial one: "Philosophical Investigations," by Ludwig Wittgenstein.
>The Old Man and the Sea," by Ernest Hemingway (hehe... but yes, I'm serious). I lol'ed
If you need any of these books, I can... you know... help.
Ah, I've forgotten. It's extremely important to know how to program. Unfortunately, this is my weakest point. If someone wants to help on this, be my guest!
i am getting a boner reading this. i humbly thank you.
I know "The Old Man and the Sea" sounds like a joke, but that novel gets you into the physics-doing mindset. It's really important. Really reallly reaaally important.
To date, my exposure to Penrose has been through his Penrose tilings. Just sayin'
Haha, I'm completely unfamiliar with Penrose. I've only read his book recently.
He did dedicate an unusual amount of his book to geometrical sets though.
Would you recommend Euclid's Elements as a place to get the basics down to pat?
I have not read it.
I feel like I typed a wall of text. Maybe I should make it more readable?
It basically sets down 5 axioms and 5 theorems based on axioms based on the geometrical nature of the universe, then comes to all geometrical knowledge that was written in his book. Would you consider it to be of practical use in my studies of physics? And no, this is perfect. I have already bookmarked this. And I rarely bookmark.
That's similar to "Principia Mathematica" by Bertrand Russell, which develops the basic logical axioms, and extends it to the fundamental theorems of every branch of modern mathematics. Since Russell is much newer, I'd suggest you go with him (he has likely improved on what Euclid did), but it may be that Euclid covers something that he hasn't.
@hkim Well Einstein was impressed with it.

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