The thin ribbons and the bulk cylindrical rods with diameters of 2 mm and 10 mm of the Vit1 metallic glass (MG) were prepared by the single roller melt spinning method and the copper mold injection casting method, respectively. The cooling rates of the samples during melt solidification were evaluated. The glass transition behaviors of three groups of MG samples with different solidification cooling rates were studied by differential scanning calorimetry (DSC) at different heating rates. The effects of melt cooling rate on the glass transition kinetic parameters such as apparent activation energy (E) and fragility parameter (m) of the Vit1 MG were studied using the Kissinger and the Vogel-Fulcher-Tammann (VFT) equations. Additionally, the structural relaxation enthalpy (ΔHrel) of three groups of MG samples was quantitatively analyzed by DSC through multi-step temperature rise and fall measurements. Results show that the melt cooling rate (R) has a significant effect on the glass transition kinetics and the structural relaxation of the Vit1 MG. As R decreases in the order of magnitude, the glass transition temperature (Tg), E, m, and ΔHrel of the Vit1 MG gradually decreases. Furthermore, in the range of the experimental cooling rates, E, m, and ΔHrel all have an approximately linear relationship with lgR.