This paper and a forthcoming one develop the spectral theory of ballooning transformations relevant to tokamak physics from first principles in a rigorous and yet intuitively clear manner. The power of the ballooning representation to throw light on the spectral characteristics of the plasma problems to which it is applicable is emphasized, and examples are given to illustrate the general notions. The ballooning representation is shown to be essentially a method to separate variables and reduce two-dimensional partial differential equations with periodic coefficients to infinite sets of soluble ordinary differential equations. This paper is concerned with an elementary approach to the techniques in the context of nearly exactly soluble problems involving the anisotropic diffusion operator in toroidal geometry. Two different perturbation methods are discussed. Applications to plasma instability problems and the subtleties involving the continuous spectra of ballooning operators will be taken up in Part 2.