The rising of topological materials MnBi2Te4/(Bi2Te3)n with built-in magnetization provides a great platform for the realization of both Chern-insulator and axion-insulator phases, manifesting integer and half-integer quantum anomalous Hall (QAH) effects, respectively. Using both model Hamiltonian and first-principles calculations, we demonstrate that rich 2D and 3D topological phase diagrams can be established with the mapping of MnBi2Te4/(Bi2Te3)n systems. For the 2D topological phases, we provide design principles to trigger integer QAH states by tuning experimentally accessible knobs, such as slab thickness and magnetization. For the 3D topological phases, we find that the surface anomalous Hall conductivity in the axion-insulator phase is a well-localized hanging around e2/2h, depending on the magnetic homogeneity. We then discuss the potential experimental signatures of the surface anomalous Hall effect in MnBi2Te4/(Bi2Te3)n.