Glasses Al2/3O-TeO2, ZnO-TeO2 and R2/3O-ZnO-TeO2 (R = Al, B) were prepared by melting in Pt crucibles and studied for correlations between structure and thermal as well as mechanical properties, whereby the glass composition is varied to tailor the short-range speciation of tellurite, aluminate, and borate groups. The glass structure was studied by Raman and infrared spectroscopy analyses, and the measured properties include glass-transition temperature (Tg), density (ρ), and ultrasonic longitudinal (VL) and transverse (VT) velocities. In addition, atomic packing density (Cg), elastic moduli, and Poisson's ratio (σ) were evaluated from the measured properties. It was found that Al2/3O leads to cross-linked alumino-tellurite networks by strong Te-O-Al bonds, which cause a profound enhancement in Tg. The influence of ZnO and B2/3O on Tg is relatively smaller due to the weaker cross-linking effects of ZnO4 tetrahedra and of Te···O-B bonds. Short-range bonding characteristics, interatomic bonding energy differences, and atomic packing density were found to have a strong effect on VT and mostly on the VL sound velocity. The combined effects of structure and bonding are nicely expressed in the composition dependence of Poisson's ratio; it exhibits decreasing trends with Al2/3O content in the binary and ternary glasses studied here, but increasing trends with ZnO and B2/3O additions in glasses ZnO-TeO2 and B2/3O-ZnO-TeO2, respectively. The results for Poisson's ratio and atomic packing density for the studied glasses were found to fit nicely in the global σ versus Cg correlation established previously for a range of glasses not including tellurites so far. Finally, the sound velocities and Poisson's ratio of pure TeO2 glass were determined for the first time and found to differ markedly from those in the literature for TeO2 glass melted in alumina crucible; this is because the latter glass is highly doped by Al2O3 leached from the alumina crucible.