This is the first paper in a series ( of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl( V, GE) and the theory of its deformations leading to metric geometric algebras Cl( V, G) and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in Cl( V, G) with easier ones in Cl( V, GE) ( e. g. a noticeable relation between the Hodge star operators associated to G and GE). Several useful examples are worked in details for the purpose of transmitting the "tricks of the trade".46927964