Gas turbine cooling system design is constrained by a maximum allowable wall temperature (dictated by the material and the life requirements of the component), minimum coolant mass flow rate (the requirement to minimize cycle-efficiency cost), and uniform wall temperature (to reduce thermal stresses). These three design requirements form the basis of an iterative design process. The relationship between the requirements has received little discussion in the literature, despite being of interest from both a theoretical and a practical viewpoint. In this paper, we consider the optimum cooling system for parts with both internal and film cooling. We show analytically that the coolant mass flow rate is minimized when the wall temperature is uniform and equal to the maximum allowable wall temperature. Thus, we show that achieving uniform wall temperature achieves minimum coolant flow rate, and vice versa. The purpose is to clarify the interplay between two design requirements that are often discussed separately in the literature. The penalty (in terms of coolant mass flow) associated with cooling nonisothermal components is quantified. We show that a typical high pressure nozzle guide vane (HPNGV) operating isothermally at the maximum allowable wall temperature requires two-thirds the coolant of a typical nonisothermal vane. The optimum coolant exit temperature is also considered. It is shown analytically that the optimum coolant exit temperature depends on the balance between the mean adiabatic film cooling effectiveness, the nondimensional mass flow rate, and the Biot number of the thermal barrier coating (TBC). For the large majority of gas turbine cooling systems (e.g., a typical HPNGV) it is shown that the optimum coolant exit temperature is equal to the local wall temperature at the point of injection. For a small minority of systems (e.g., long effusion cooling systems operating at low mass flow rates), it is shown that the coolant exit temperature should be minimized. An approximation relating the wall/film properties, the nondimensional mass flow, and the overall cooling effectiveness is derived. It is used to estimate the effect of Biot number (TBC and metal), heat transfer coefficient (HTC) ratio, and film properties on the performance of a typical HPNGV and effusion cooling system. In Part II, we show that designs which achieve uniform wall temperature have a particular corresponding internal HTC distribution.