We investigate how different chemical environments influence magnetic properties of terbium(III) (Tb)-based single-molecule magnets (SMMs), using first-principles relativistic multireference methods. Recent experiments showed that Tb-based SMMs can have exceptionally large magnetic anisotropy and that they can be used for experimental realization of quantum information applications, with a judicious choice of chemical environment. Here, we perform complete active space self-consistent field calculations including relativistic spin-orbit interaction for representative Tb-based SMMs such as TbPc2 and TbPcNc in three charge states. We calculate the low-energy electronic structure from which we compute the Tb crystal-field (CF) parameters and construct an effective pseudospin Hamiltonian. Our calculations show that the ligand type and fine points of molecular geometry do not affect the gap between the ground-state and first-excited doublets, whereas the latter varies weakly with oxidation number. On the other hand, higher-energy levels have a strong dependence on all these characteristics. For neutral TbPc2 and TbPcNc molecules, the Tb magnetic moment and ligand spin are parallel to each other and the coupling strength between them does not depend much on the ligand type and details of the atomic structure. However, ligand distortion and molecular symmetry play a crucial role in transverse CF parameters which lead to tunnel splitting. The tunnel splitting induces quantum tunneling of magnetization by itself or by combining with other processes. Our results provide insights into the mechanisms of magnetization relaxation in the representative Tb-based SMMs.