The purpose of this article is to introduce the notion of constructing any arbitrary finite and infinite types of non-compact hyperbolic Riemann surfaces via (non-abelian) fundamental groups equipped with various types of classical Schottky structures, with limit sets as uncountable sets (but not necessarily Cantor sets), emphasising the cases in which the surfaces are of infinite hyperbolic areas. In particular, in this paper, the primary goal is to establish the existence of a canonical non-compact infinite area Fuchsian polygon with the help of various classical Schottky structures in the hyperbolic plane. After that, we have initiated a structure of an arbitrary finite type non-compact hyperbolic Riemann surface with genus, conformal holes, cusps, and funnel ends by using the canonical Fuchsian Schottky polygons. Furthermore, in this manuscript, we have proposed the ideas of infinite types conformally compact and semi-conformally compact hyperbolic Riemann surfaces, respectively. Indeed, we have constructed four new and interesting types of infinite type hyperbolic Riemann surfaces (we call generalized flute surfaces) that are constructed from infinite sequences of infinite area hyperbolic pair of pants, each glued to the next along a common geodesic boundary with certain methodologies.