Today, I flipped through Halliday's Fundamentals of Physics, to get a sense of what topics would make for a good starting place in physics. I'm starting this list with Newtonian mechanics: particle movement and interaction, and gravity. Throwing in the continuum approximation -- the notion that a substance made of a large number of small interacting particles acts mathematically as if it were made of a continuous material -- allows for a discussion of fluid mechanics. Also, it might be useful to go over oscillation, as the simple harmonic oscillator -- imagine a weight attached to a spring, moving back and forth -- is an example used repeatedly in the quantum mechanics books I've looked through. After Newtonian mechanics is basic wave theory, then thermodynamics and the study of gases, and electromagnetics (culminating in Maxwell's laws). Finally, there is optics. Mathematically, I would say a familiarity with vector calculus would be sufficient; but upon reaching optics, at least on the introductory level, I don't see why much math beyond trigonometry would be needed.
This would be the usual first-year mathematical treatment of the science. However, there is another way to approach physics mathematically, and for anyone interested in progressing to quantum mechanics, I would recommend that after they try the above treatment, they try this method. In physics, many objects behave in a way which minimizes or maximizes some value; for example, consider Fermat's principle, that light travels along the path which takes the least amount of time to traverse. Thus, to find the path that light travels under certain conditions, one must find the curve which minimizes the time needed to traverse under those conditions. The mathematics of optimizing over functions is known as the calculus of variations.
Rounding off this post, I turn to a topic I wish someone would have told me about sooner: statistical mechanics. In thermodynamics, one starts with three or four laws, and uses these to describe how systems share energy. Statistical mechanics shows how the laws of thermodynamics arise from considering a system as a collection of many tiny particles. Granted, it ignores the properties of different materials by locking them away in various constants, so it doesn't answer everything about why materials behave the way they do. (In other words, as I was recently chastened: thermodynamics is not about information, but about the transfer of energy.) But statistical mechanics does go a ways toward that goal. Statistical mechanics requires a simple introduction to probability. Also, a quick note for those who decide to use the MIT OpenCourseWare notes: the statistical mechanics course is listed under physics, but thermodynamics, given its importance to understanding chemical kinetics, is listed under chemistry.
So that's it for now. I'll probably post soon about quantum mechanics and relativity, and maybe have a post on mathematical biology. Also, in the next week or so, I'll try to gather up some useful links relating to today's post.