The phase behavior of confined fluids is rich even for simple models of fluids and simple confining geometries. There has been a great deal of work to understand these systems, and density functional theories (DFT) of inhomogeneous fluids are often applied to determine phase diagrams quickly for these simple systems where symmetry in the physical problem reduces the computational problem to a one-dimensional calculation. More recently, there has been interest in developing DFT algorithms for treating fluids in complex confining geometries or at chemically heterogeneous surfaces where two- or three-dimensional calculations are required. In this paper we present three algorithms for the rapid and robust study of phase behavior in DFT models of inhomogeneous fluids and demonstrate their utility by analyzing capillary condensation in slit pores and ordered two-dimensional arrays of cylindrical fibers. The three algorithms are arclength continuation algorithms for tracing connected stable, metastable, and unstable branches, a phase transition tracking algorithm that allows for rapid computation of phase envelopes, and a spinodal tracking algorithm that allows one to assess the limits of metastability of a given state. In Paper II of this series, we apply these algorithms in a detailed investigation of capillary condensation in disordered porous media. (C) 2003 American Institute of Physics.