How about the study of minimal surfaces (physical applications in soap films etc.)? In fact one might argue the Lagrangian formulation of minimal surfaces (the problem of Plateau) is one of the oldest "classical field theory" problems, and led to the revival of calculus of variations in the early twentieth century (see esp. the works of Morrey).
Slightly related is the general study of continuum mechanics and (non-linear) elasticity. Which is kind of like fluid mechanics except for deformations of solids.
Another well-known application of the general frame work is the study of harmonic maps and wave maps (also known as non-linear sigma model in physics). The study of such systems led to developments of the techniques of compensated compactness and multilinear product estimates in partial differential equations (see, e.g. works of Helein, Klainerman, Tao, Krieger, and many others). The regularity properties of the harmonic maps are still under active study (Li and Tian, Nguyen, Weinstein, and others). And in physics, the sigma models find application from particle physics (as a model for equivariant Yang-Mills equation) to general relativity (stationary solutions in Einstein-vacuum or Einstein-Maxwell theories).
The sigma models are also generalized by Tony Skyrme in his namesake quasilinear model (both hyperbolic and elliptic), which is not yet well understood. This model has found applications from nucleon physics to condensed matter, and now to topological material science. The study of the stationary problem (and its generalization in the Fadeev-Skyrme model) led to interesting developments in topology and geometry (since the model admits topological solitons), see for example the work of Kapitansky.
If you really know nothing about physics I suggest you begin with any text book on physics for undergrad. Easy to read, it will introduce the main usual suspects. After, you'll ask again :)
I am not sure that jumping from nothing to quantum mechanics, or even worse quantum fields theory, would be wise, like jumping from nothing in math to algebraic geometry or K-Theory.
After that, it depends of course at what level of mathematical physics you want to stop. I will illustrate this with some examples:
Question: What is the "mass" of an isolated dynamical system?
Math Answer: It is the class of cohomology of the action of the group of Galilee, measuring the lack of equivariance of the moment map, on a symplectic manifold representing the isolated dynamical system.
Another question: Why in general relativity $E = mc^2$?
Math Answer: Because the group of Poincaré has no cohomology
Another, other question: What is the theorem of decomposition of motions around the center of gravity?
Math Answer: Let $(M,\omega)$ be a symplectic manifold with an hamiltonian action of the group of Galilee, if the "mass" of the system is not zero (in the sense above) then $M$ is the symplectic product or $({\bf R}^6, {\rm can})$, representing the motions of the center of gravity, by another symplectic manifold $(M_0,\omega_0)$, representing the motions around the center of gravity. The group of Galillee acting naturally on $\bf R^6$ and $SO(3) \times {\bf R}$ on $M_0$.
Another, other, other question: What are the constants of motions?
Math Answer: Let $(M,\omega)$ be a pre-symplectic manifold with an hamiltonian action of a Lie group $G$, then the moment map is constant on the characteristics of $\omega$, that is the integral manifolds of the vector distribution $x \mapsto \ker(\omega_x)$.
These answers are the mathematical versions of physics classical constructions, but it would be very difficult to appreciate them if you have no pedestrian introduction of physics. You may enjoy also Aristotles' book "Physics", as a first dish, just for tasting the flavor of physics :)
After that, you will be able to appreciate also quantum mechanics, but this is another question.
Addendum
Just before entering in the modern world of physics I would suggest few basic lectures for the winter evenings, near the fireplace (I'm sorry I write them down in french because I read them in french).
• Platon, Timée, trad. Émile Chambry.
• Aristote, La Physique, Éd. J. Vrin.
• Maïmonide, Le Guide des Égarés, Éd. Maisonneuve & Larose. (the part about time as an accident of motion, accident of the thing. Very deep and modern thoughts).
• Giordano Bruno, Le Banquet des Cendres, Éd. L’éclat.
• Galileo Galilei, Dialogue sur les Deux Grands Systèmes du Monde, Éd. Points.
• Albert Einstein, La Relativité, Éd. Payot.
• Joseph-Louis Lagrange, Mécanique Analytique, Éd. Blanchard.
• Felix Klein, Le Programme d’Erlangen, Éd. Gauthier-Villars.
• Jean-Marie Souriau, Structure des Systèmes Dynamiques, Éd. Dunod.
• Victor Guillement & Shlomo Sternberg, Geometric Asymptotics, AMS Math Books
• François DeGandt Force and Geometry in Newton Principia.
Best Answer
The label "geometic algebra" was William Cifford's name for the algebra he discovered (invented). This bit of history is recounted by David Hestenes, who has perhaps been most responsible for promoting this lovely mathematics for physics, and has many books and papers on this subject. There is also a Cambridge lot, including Chris Doran and Anthony Lasenby, who have a physics book, and a gauge theory of gravity. These men are perhaps the most influential authors, but I would also want to include Perti Lounesto, and William Baylis, in the geometric algebra crowd. I have personally found these authors particularly accessible and a pleasurable learning experience.
What seems to me the most unifying element of those who use the term "geometric algebra" for "Clifford algebra" is the emphasis on a real exposition of the subject. To my mind, since complex numbers are in fact themselves a real geometric algebra, there is no small bit of confusion generated by insisting on the mathematical development of the subject over a complex field. But this is exactly what Emile Cartan did in defining spinors. With spinors deeply embedded in physical theories of elementary particles, a Clifford algebra defined over a complex field has a strong tradition, especially among mathematically inclined theorists. Most algebraists (Chevalley, Cartan, Atiyah) would consider complex numbers the truest form of 'number'.
Before closing on the subject, a serious researcher should include a third subject: "noncommutative algebra". In many respects, this is a similar development but with a pedigree more after Hermann Grassmann than William Clifford. Alain Connes, has developed this subject recently with physical applications.
In conclusion, I offer the advice that 'geometric algebra' is the most accessible and intuitive approach offering a physically descriptive mathematics. It is just multivariate linear algebra from a practical vantage point. I think it should be taught in high school. The 'Clifford algebra' and 'noncommutative algebra' approaches are more abstract and mathematically rigorous, but greatly expand the available literature when used as search terms.