The magnetic van der Waals crystals MnBi_{2}Te_{4}/(Bi_{2}Te_{3})_{n} have drawn significant attention due to their rich topological properties and the tunability by external magnetic field. Although the MnBi_{2}Te_{4}/(Bi_{2}Te_{3})_{n} family have been intensively studied in the past few years, their close relatives, the MnSb_{2}Te_{4}/(Sb_{2}Te_{3})_{n} family, remain much less explored. In this work, combining magnetotransport measurements, angle-resolved photoemission spectroscopy, and first principles calculations, we find that MnSb_{4}Te_{7}, the n=1 member of the MnSb_{2}Te_{4}/(Sb_{2}Te_{3})_{n} family, is a magnetic topological system with versatile topological phases that can be manipulated by both carrier doping and magnetic field. Our calculations unveil that its A-type antiferromagnetic (AFM) ground state stays in a Z_{2} AFM topological insulator phase, which can be converted to an inversion-symmetry-protected axion insulator phase when in the ferromagnetic (FM) state. Moreover, when this system in the FM phase is slightly carrier doped on either the electron or hole side, it becomes a Weyl semimetal with multiple Weyl nodes in the highest valence bands and lowest conduction bands, which are manifested by the measured notable anomalous Hall effect. Our work thus introduces a new magnetic topological material with different topological phases that are highly tunable by carrier doping or magnetic field.