The antiferromagnetic semiconductor MnTe has recently attracted attention for spintronics and high-performance thermoelectric applications. However, little is known about its vibrational and thermal transport properties and how these might relate to the electronic and magnetic structure, particularly as related to 3d Mn orbital correlations. Here, we calculate a physically justified Coulomb correlation parameter within the DFT+U framework. We couple this framework with the Heisenberg Hamiltonian and first-principles Boltzmann transport to understand the magnetic, vibrational, and phonon thermal transport properties of MnTe. We also perform inelastic neutron and nuclear inelastic x-ray scattering measurements of the total and partial phonon density of states, respectively. Very good agreement is obtained with the measured and calculated phonon density of states, and with available measurements for the band gap, local magnetic moments, Néel temperature, magnon dispersion, thermal conductivity, and phonon dispersion. This study demonstrates that the vibrational and magnetic degrees of freedom are not strongly coupled in MnTe, and provides a more comprehensive picture of this technologically promising material.