This is the first of two papers on the propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. We first show that the nonlinear system of equations governing the plasma and the electromagnetic field in a given, external gravitational field has locally a unique solution for any initial data set obeying the appropriate constraints, and that this system is linearization stable at any of its solutions. Next we prove that the linearized perturbations of a `background' solution are characterized by a third-order (not strictly) hyperbolic, constraint-free system of three partial differential equations for three unknown functions of the four space-time coordinates. We generalize the algorithm for obtaining oscillatory asymptotic solutions of linear systems of partial differential equations of arbitrary order, depending polynomially on a small parameter such that it applies to the previously established perturbation equation when the latter is rewritten in terms of dimensionless variables and a small scale ratio. For hyperbolic systems we then state a sufficient condition in order that asymptotic solutions of finite order, constructed as usual by means of a Hamiltonian system of ordinary differential equations for the characteristic strips and a system of transport equations determining the propagation of the amplitudes along the rays, indeed approximate solutions of the system. The procedure is a special case of a two-scale method, suitable for describing the propagation of locally approximately plane, monochromatic waves through a dispersive, inhomogeneous medium. In the second part we shall apply the general method to the perturbation equation referred to above.