Using first principles methods, we systematically study the crucial steps of CVD growth of CNTs, the binding and the diffusion of the carbon feedstock as well as the nucleation of CNTs. Late transition (Ni, Pd, Pt) and coinage (Cu, Ag, Au) metal catalysts were investigated. For all metals, we considered various diffusion mechanisms including both surface and subsurface channels, finding the lowest activation barriers for carbon adatoms on nanoparticles of coinage metals. For these metals, our calculations further show that the diffusion is restricted to the nanoparticle surface when diatomic carbon is initially obtained from the decomposition of the precursor gas. From the binding energies of armchair and zigzag edges of CNT fragments, we infer a high preference for the growth of armchair CNTs. These results indicate that coinage metal catalysts, in particular Cu, favor CVD growth of CNTs at low temperatures and with narrow chirality distributions.