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This three-part paper describes linear and nonlinear plasma wave phenomena in an infinite, homogeneous, collisionless, warm magnetoplasma by means of a microscopic Lagrangian. Part 1 derives the dispersion relation for all modes of linear wave propagation. To do so, the charged particle position vectors and the fields are first expanded in terms of sinusoidal perturbations from equilibrium. The contribution to the microscopic Lagrangian, expanded to second order in the perturbations, is then averaged over space and time to remove rapidly varying terms. The Euler-Lagrange equations, obtained from variations of the Lagrangian with respect to the amplitudes of the perturbation parameters, are the first-order Maxwell equations and the perturbed particle trajectory. Variation with respect to the phase gives the equation of conservation of action. The Lagrangian is specialized to waves propagating nearly parallel, and exactly perpendicular, to the static magnetic field, and familiar wave dispersion relations are obtained. In Part 2, the nonlinear coupling of these waves is studied. In Part 3, both wave-wave and wave-particle interactions are taken into account.