Kagome-lattice magnets RMn6Sn6 recently emerged as a new platform to exploit the interplay between magnetism and topological electronic states. Using ab initio methods, we systematically investigate the electronic structures and intrinsic magnetic properties in RMn6Sn6 with R = Gd, Tb, Dy, Ho, and Er. We reveal a non-monotonic dependence of the magnetocrystalline anisotropy energy on the spin quantization for all RMn6Sn6, except for R = Gd. Moreover, the Mn sublattice exhibits easy-plane anisotropy of similar amplitude in all RMn6Sn6 compounds. Our results show that TbMn6Sn6 has an easy-axis anisotropy, while DyMn6Sn6 and HoMn6Sn6 have easy-cone anisotropy, and GdMn6Sn6 and ErMn6Sn6 have easy-plane anisotropy. These numerical findings all agree well with the experiment and are fully consistent with the Mn coordination of the R atoms. The observed band structures of various RMn6Sn6 compounds share great similarities near the Fermi level as they mainly consist of non-4f bands. Multiple Dirac crossings occur at the Brillouin zone corners, opened by spin-orbit coupling (SOC). Most of these crossings are strongly kz-dependent. The most prominent 2D-like Dirac crossing that can be effectively gapped by SOC lies ∼ 0.7 eV above the Fermi level, much higher than in previously reported band structures calculated in density functional theory [Ref. 1]. The inclusion of electron correlations within Mn-3d electrons, however, can significantly lower this SOC-gapped crossing closer to the Fermi level, making it more relevant for the observation of topological phenomena. Finally, we predict an enhancement of the Mn magnetic moment on the RMn6Sn6 surface, which may impact the topological band structures of surfaces or thin films, and is yet to be confirmed experimentally.