The central theme in this series of papers is the development of a coherent and consistent equation-of-state model of fluids in the frame of solvation thermodynamics. Two main ingredients will be used for this purpose: the first will be the successful conductor-like screening model (COSMO) type group-contribution models with their quantum-mechanical description of fluids, and the second, our recent nonrandom hydrogen-bonding (NRHB) equation-of-state framework. The bridge between the two is the quasi-chemical treatment of the nonrandom distribution of molecular entities in the system, a treatment used by Henry Kehiaian in his DISQUAC group-contribution model. In this first part we develop the solvation formalism of NRHB and propose a first approach for turning the COSMO model into an equation-of-state model in a straightforward and consistent manner. Although the formalism is general, the focus is on the self-solvation of fluids. This development permits the calculation of the various components of solvation Gibbs energy, such as the cavitation and charging components or the enthalpic and the entropic ones. It permits, in particular, their study over a broad range of external conditions. The strength of the intermolecular forces and its role on the variation of these components, as well as their interdependence, are critically discussed.