We study the conductance through a single vibrating molecule contacted to two metallic electrodes. The starting point is a tight-binding Hamiltonian with a linear coupling of the ionic motion to the electronic degrees of freedom. We distinguish between a coupling to the onsite energy of the molecule and a coupling representing a bond-stretching vibrational mode. The full nonequilibrium current, shot noise, and dissipated power is calculated within the Keldysh Green function formalism. As far as transport observables are concerned, onsite and bond-stretching coupling mechanisms differ in several respects: In the latter case satellite peaks in the differential conductance are more pronounced and results closer to the experimental evidence can be obtained.