In Part I of a series of study on Lagrangian-conic relaxations, we introduce a unified framework for conic and Lagrangian-conic relaxations of quadratic optimization problems (QOPs) and polynomial optimization problems (POPs). The framework is constructed with a linear conic optimization problem (COP) in a finite dimensional vector space endowed with an inner product, where the cone used is not necessarily convex. By imposing a copositive condition on the COP, we establish fundamental theoretical results for the COP, its conic relaxations, its Lagrangian-conic relaxations, and their duals. A linearly constrained QOP with complementarity constraints and a general POP can be reduced to the COP satisfying the copositivity condition. Then, the conic and Lagrangian-conic relaxations of such a QOP and POP are discussed in a unified manner. The Lagrangian-conic relaxation takes one of the simplest forms, which is very useful to design efficient numerical methods. As for applications of the framework, we discuss the completely positive programming relaxation, and a sparse doubly nonnegative relaxation for a linearly constrained QOP with complementarity constraints. The unified framework is applied to general POPs in Part II.