The Gilson-Pickering equation is a nonlinear partial differential equation that models complex wave propagation, incorporating nonlinear steepening, higher-order dispersion, and mixed derivatives. This paper constructs new exact solutions using analytical methods, focusing on the Hirota bilinear method, which is used to bilinearize the equation, facilitating the derivation of explicit solutions. Rational lump solutions are derived, examining their localized and algebraically decaying properties, providing insight into wave interaction dynamics. Breather solutions are also obtained, capturing oscillatory and time-periodic behavior, highlighting synchronization effects, bound-state formation, and complex collision phenomena. The solutions are analyzed within their mathematical framework, emphasizing distinctive features and physical significance. The results enhance the understanding of nonlinear wave dynamics governed by the Gilson-Pickering equation, offering a foundation for further theoretical and applied investigations of higher-order multidimensional nonlinear models. The study sheds light on intricate dynamics of nonlinear waves, enabling better comprehension of complex wave propagation phenomena in various physical systems, with potential applications in fields like fluid dynamics, optics, and plasma physics. This research opens up new directions in nonlinear science and applied mathematics, enhancing our understanding of complex nonlinear systems and their behavior, and creating many opportunities for future studies and exploration across diverse scientific and engineering applications worldwide.