Cobalt thin films with 5 nm thickness were prepared on single-crystal MgO (001) substrates with different thickness Cu buffer (0 nm, 5 nm, 10 nm, 20 nm). The structure, magnetic properties and transport behaviors were investigated by employing low-energy-electron-diffraction (LEED), magneto-optical Kerr effect (MOKE) and anisotropic magnetoresistance (AMR). By comparing the magnetic properties of the sample as-deposited (without Cu buffer layer) one with those having the buffer Cu, we found that the magnetic anisotropy was extremely affected by the Cu buffer layer. The magnetic anisotropy of the as-deposited, without buffer layer, sample shows the uniaxial magnetic anisotropy (UMA). We found that the symmetry of the magnetic anisotropy is changed from UMA to four-fold when the thickness of the Cu buffer layer reaches to 20 nm. Meanwhile, the coercivity increased from 49 Oe (without buffer layer) to 300 Oe (with 20 nm Cu buffer), in the easy axis direction, as the thickness of the buffer layer increases. Moreover, the magnitudes of various magnetic anisotropy constants were determined from torque curves on the basis of AMR results. These results support the phenomenon shown in the MOKE.