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Using ab initio methods and atomic spin dynamics, we investigate the electronic structure and magnetism in MnBi₂Te₄ and MnSb₂Te₄. By comparing the bandstructure obtained using DFT+U and Quasi-particle self-consistent GW methods, we found that a U≈ 5 eV Hubbard correction on Mn-d orbitals is needed to describe the electronic structure and magnetic ordering better with DFT+U. We also investigate the intrinsic magnetic properties in these compounds, including exchange couplings and magnetocrystalline anisotropy. The anti-site defects are found to form more easily in MnSb₂Te₄ than in MnBi₂Te₄, creating antiferromagnetic Mn-Mn couplings within the septuple block. We investigate the origin of anisotropy and the effects of dimensionality, anti-site defects, and interlayer ordering on anisotropy. Our calculated results are in reasonable agreement with the recent experiments. We also discuss the origin of non-hysteretic spin-flop behavior in these compounds.