We study the electric and thermal transport of the Dirac carriers in monolayer graphene using the Boltzmann-equation approach. Motivated by recent thermopower measurements [F. Ghahari, H.-Y. Xie, T. Taniguchi, K. Watanabe, M. S. Foster, and P. Kim, Phys. Rev. Lett. 116, 136802 (2016)], we consider the effects of quenched disorder, Coulomb interactions, and electron–optical-phonon scattering. Via an unbiased numerical solution to the Boltzmann equation we calculate the electrical conductivity, thermopower, and electronic component of the thermal conductivity, and discuss the validity of Mott's formula and of the Wiedemann-Franz law. An analytical solution for the disorder-only case shows that screened Coulomb impurity scattering, although elastic, violates the Wiedemann-Franz law even at low temperature. For the combination of carrier-carrier Coulomb and short-ranged impurity scattering, we observe the crossover from the interaction-limited (hydrodynamic) regime to the disorder-limited (Fermi-liquid) regime. In the former, the thermopower and the thermal conductivity follow the results anticipated by the relativistic hydrodynamic theory. On the other hand, we find that optical phonons become non-negligible at relatively low temperatures and that the induced electron thermopower violates Mott's formula. Combining all of these scattering mechanisms, we obtain the thermopower that quantitatively coincides with the experimental data.