We present a systematic method for constructing static, spherically symmetric regular spacetimes of Kerr-Schild type in general relativity that satisfy the weak energy condition. Our approach relies on physically reasonable assumptions on the matter energy density, together with the boundedness of the Kretschmann scalar. The latter property ensures the finiteness of all curvature invariants and, for the configurations considered, is equivalent to the completeness of causal geodesics. By classifying admissible density profiles according to their complexity, we recover well-known regular black hole solutions such as the Bardeen, Hayward, and Dymnikova models, which are thus naturally embedded in a unified and broader framework. Within this setting, we also derive closed-form expressions for several new families of regular geometries involving hypergeometric or incomplete Gamma functions, which in many cases reduce to elementary functions including algebraic, logarithmic, arctangent, and exponential forms. The emergence of horizons and photon spheres, as well as their observational implications, is also investigated. Finally, we extend the analysis to rotating metrics generated through the Newman-Janis algorithm.