I've found some nice sources of information and clear explanations during my decade of studying physics. I wish someone had told me about these resources to begin with. This page contains some advice that I would have given myself when I was beginning my college career.

General Recommendations

Schaum's Outlines

Schaum's outlines are an excellent resource for learning the mathematics required for physics majors and graduate students. They are inexpensive, they clearly summarize the material, and they contain many detailed solutions to problems. Many of the books are sufficient to use as a primary source, however the best approach is to have at least one traditional textbook on the subject and one Schaum's Outline if you are teaching yourself. I especially recommend the following:

Complex Variables Advanced Calculus Linear Algebra Differential Equations Partial Differential Equations Tensor Calculus Fourier Analysis Differential Geometry

The Feynman Lectures on Physics

This three-volume set is very rich with clear, concise explanations. It covers practically the entire core physics curriculum including a more thorough discussion of fluid dynamics than most undergraduate physics programs in the United States offer. I highly recommend this to anyone studying physics at the undergraduate or graduate level. Refer to the Wikipedia article for more information: The Feynman Lectures on Physics.

Landau & Lifshitz Course of Theoretical Physics

This series of ten texts covers the core of graduate-level physics. Lev Landau---like his American counterpart, Feynman---was very talented at explaining things.

vol. 1: Mechanics vol. 2: The Classical Theory of Fields vol. 3: Quantum Mechanics: Non-Relativistic Theory vol. 4: Quantum Electrodynamics vol. 5: Statistical Physics Pt. 1 vol. 6: Fluid Mechanics vol. 7: Theory of Elasticity vol. 8: Electrodynamics of Continuous Media vol. 9: Statistical Physics Pt. 2 vol. 10: Physical Kinetics

Inspiration & Motivation

Studying physics at a university can be arduous and boring at times, so it helps to read some popularizations to remind yourself that physics is fun and exciting once you learn enough. Reading this sort of material can also help you to explain things to people who aren't as interested in physics as you are. Here are a few examples:

Surely You're Joking, Mr. Feynman! - Richard P. Feynman QED - Richard P. Feynman The Emperor's New Mind - Sir Roger Penrose Relativity - Albert Einstein The Elegant Universe - Brian Greene

Specific Recommendations

Fluid Dynamics

I strongly encourage people to study fluid dynamics because it makes vector and tensor calculus more intuitive. Fluids are found everywhere around us and inside of us, so it's nice to have a good understanding of the general properties of fluid dynamics. Once you understand fluids, you will find that many aspects of electricity & magnetism become intuitive. In addition to volume 2, chapters 39 - 41 of the Feynman Lectures and volume 6 of Landau & Lifshitz, I recommend the excellent text by Batchelor:

An Introduction to Fluid Dynamics - G.K. Batchelor

Classical Mechanics

In this case, the standard text by Goldstein already covers the subject very well. If it is supplemented with volume 1 of the Lifshitz & Landau series and volume 2, chapter 19 of the Feynman lectures, you will be likely to have a more solid understanding of the subject. I would recommend studying graduate-level mechanics after or at the same time that you study tensor calculus.

Electricity & Magnetism / Classical Electrodynamics

As I mentioned above, I recommend studying fluids before electromagnetic theory. In addition to volume 2 of the Feynman Lectures, volumes 2 and 8 of Landau & Lifshitz, and the standard text by Jackson, I highly recommend:

Electricity and Magnetism - Oleg D. Jefimenko Classical Electromagnetism - Jerrold Franklin Classical Electricity & Magnetism - Panolfsky & Phillips

The text by Jefimenko is very impressive. It contains many clear examples and explanations. The Franklin text contains a nice explanation of shortcuts that can be used when manipulating differental vector operators and it avoids SI units, so it doesn't contain random $\epsilon_0$'s and $\mu_0$'s. Thus it's more in line with research papers. The newest edition of Jackson's text uses the SI system in part of the book and then switches to Gaussian units later when it gets to more advanced topics;  it's not self-consistent. The text by Panolfsky and Phillips is a classic that contains a few derivations that are otherwise hard to find; some people consider it the best book on the subject.

Many people are fans of the undergraduate text by David Griffiths.  I can't comment on that particular text since I haven't read it. The text by Julian Schwinger et al. is probably the most advanced / mathematical graduate text on the subject

Tensor Calculus

I recommend using a combination of the following two books to begin with:

Introduction to Tensor Calculus and Continuum Mechanics - John H. Heinbockel Shaum's Outline of Tensor Calculus - David Kay

These will introduce you to all of the essential details of the tensor calculus using index notation. They will also provide you with exposure to the two common notations for Christoffel symbols and they include many solved practice problems. To learn more about index-free notation and applications of tensors, it would probably be best to use a modern general relativity or differential geometry text. For the simplest application, study the stress tensor of fluid dynamics.  For an explanation of the difference between covariant and contravariant quantities, I've written this page.

Quantum Mechanics

Quantum theory is best understood once you have studied the Lagrangian and Hamiltonian formulations of classical mechanics.  Having a solid background in the solutions of the partial differential equations appearing in electrostatics is also very helpful.  In addition to the third volume of the Feynman Lectures and the third volume of Landau & Lifshitz, one book stands out as being particularly good as a primary text:

Principles of Quantum Mechanics - Ramamurti Shankar

It should be supplemented with extra material from other texts, of course. That's always the case. A very interesting and clear introduction to the path integral formulation can be found in Quantum Mechanics and Path Integrals by Feynman and Hibbs.

Thermodynamics and Statistical Mechanics

In terms of thermodynamics, the book by Fermi provides a very nice introduction to the theory:

Thermodynamics - Enrico Fermi

I haven't read an exceptional statistical mechanics text yet. A few have a pretty good reputation, so they are on my reading list:

Statistical Mechanics: A Set Of Lectures - Richard P. Feynman The Principles of Statistical Mechanics - Richard C. Tolman Nonequilibrium Statistical Mechanics - Robert Zwanzig Statistical Physics of Particles - Mehran Kardar Statistical Physics of Fields - Mehran Kardar A Modern Course in Statistical Physics - Linda E. Reichl

Recommended Order:

As far as I can tell, the best order in which to study graduate-level physics would be roughly the following. I've organized the material into five phases where the topics in each phase are studied simultaneously:

1. mechanics, fluid dynamics and thermodynamics, mathematics (tensor calculus, complex analysis)
2. more mechanics, electricity & magnetism, special relativity, more mathematics (PDE's, Green's functions, "special" orthogonal functions)
3. more electricity and magnetism (which includes more about special relativity), introductory general relativity, and quantum mechanics
4. more quantum mechanics, statistical mechanics, more general relativity, and more mathematics (differential geometry)
5. quantum field theory, elementary particles, more statistical mechanics, more mathematics (group theory and topology)

Computational physics and basic computer science can be introduced at any point.