Understanding and manipulating the interactions between foreign bodies and cell membranes during endo- and phagocytosis is of paramount importance, not only for the fate of living cells but also for numerous biomedical applications. This study aims to elucidate the role of variables such as anisotropic particle shape, curvature, orientation, membrane tension, and adhesive strength in this essential process, using a minimal experimental biomimetic system comprising giant unilamellar vesicles and rod-like particles with different curvatures and aspect ratios. We find that the particle wrapping process is dictated by the balance between the elastic energy penalty and adhesion energy gain, leading to two distinct engulfment pathways, tip-first and side-first, emphasizing the significance of the particle orientation in determining the pathway. Moreover, our experimental results are consistent with theoretical predictions in a state diagram, showcasing how to control the wrapping pathway from surfing to partial to complete wrapping by the interplay between membrane tension and adhesive strength. At moderate particle concentrations, we observed the formation of rod clusters, which exhibited cooperative and sequential wrapping. Our study not only contributes to a comprehensive understanding of the mechanistic intricacies of endocytosis by highlighting how the interplay between the anisotropic particle shape, curvature, orientation, membrane tension, and adhesive strength can influence the engulfment pathway but also provides a foundational base for future research in the field.