Interactions of graphene with 6H-SiC {0001} surfaces are numerically investigated from first principles. In order to describe the bulk structure and its 6 bilayer thick surfaces correctly, bare and dipole-corrected atomic relaxations are considered. The obtained lattice parameters and bulk modulus are in good agreement with experimental values. The calculated indirect band gap width of 2.10 eV is smaller than the experimental value due to the nature of the computational method. Geometrical optimization of the surfaces, where dipole correction is applied, reveals that the first two bilayers displace significantly, where the relaxations of the very top bilayer is more pronounced. Band structures of the {0001} surfaces possess two flat bands around the Fermi level due to unsaturated bonds on opposite faces. When one layer of C atoms are introduced on the Si-terminated surface, it behaves as a tightly-bound buffer layer. This is also the case for the C-terminated surface when van der Waals interactions are taken into account. In contrast, disregarding these interactions yields free-standing graphene like behavior for the first C overlayer. On both surfaces, the second C overlayer is free-standing where the corresponding band structures incorporate Dirac-cone like features.