The role of electron-electron interactions in two-dimensional Dirac fermion systems remains enigmatic. Using a combination of nonperturbative numerical and analytical techniques that incorporate both the contact and long-range parts of the Coulomb interaction, we identify the two previously discussed regimes: a Gross-Neveu transition to a strongly correlated Mott insulator and a semimetallic state with a logarithmically diverging Fermi velocity accurately described by the random phase approximation. We predict that experimental realizations of Dirac fermions span this crossover and that this determines whether the Fermi velocity is increased or decreased by interactions. We explain several long-standing mysteries, including why the observed Fermi velocity in graphene is consistently about 20% larger than values obtained from ab initio calculations and why graphene on different substrates shows different behaviors.