Under compressive and shear pressures, composite laminates, which are crucial load-bearing components in airplanes, are vulnerable to buckling or overstress failure. In this paper, the theoretical analysis methods for buckling and first-ply-failure were adopted to predict the failure of composite laminated plates. The investigation was carried out on graphite/epoxy plates, with different load ratios, side length ratios, side length-thickness ratios, and lay-up angles. Two failure categories’ occurrence sequence as well as the failure loads’ variance were found. It is revealed that buckling and first-ply-failure occur in different orders under various load ratios. The load ratio ranges corresponding to buckling and first-ply-failure are affected by the side length ratio, side length-thickness ratio, and lay-up angle of the plate. The buckling mode is primarily influenced by the load ratio, side length ratio and lay-up angle, while the first-layer-failure mode is mainly affected by the load ratio and lay-up angle.