We have generated precise values for several key energies that are relevant to Cu-graphene or Cu-graphite systems. Such systems may find technological applications that range from graphene synthesis, to condensation heat transfer, to electrical contacts to graphene, to composites. Using density functional theory, we have calculated surface energies of the three low-index faces of bulk Cu. We find that these surface energies, calculated with the PBEsol functional, are significantly higher than with the more common PBE functional and agree more closely with experiment. We have also calculated the surface energies of graphene and graphite, the exfoliation energy between graphene and graphite, and the adhesion energies between graphene or graphite and a Cu(111) slab. The adhesion energy between a carbon layer and Cu(111) is close to the exfoliation energy and cleavage energy of graphite, the four sets of values spanning a range of only 0.394 to 0.456 J/m2. Our results are consistent with the earlier experimental observation of three-dimensional growth of Cu on top of graphite. The energies are also used to perform a continuum Winterbottom analysis and also discrete (atomistic) variants to predict the equilibrium shapes of Cu particles supported on graphite.