We illustrate the formation and evolution of the Milky Way over cosmic time, utilizing a sample of 10 million red giant stars with full chemodynamical information, including metallicities and $\alpha$-abundances from low-resolution Gaia XP spectra. The evolution of angular momentum as a function of metallicity - a rough proxy for stellar age, particularly for high-[$\alpha$/Fe] stars - displays three distinct phases: the disordered and chaotic protogalaxy, the kinematically-hot old disk, and the kinematically-cold young disk. The old high-$\alpha$ disk starts at [Fe/H] $\approx -1.0$, 'spinning up' from the nascent protogalaxy, and then exhibits a smooth 'cooldown' toward more ordered and circular orbits at higher metallicities. The young low-$\alpha$ disk is kinematically cold throughout its metallicity range, with its observed properties modulated by a strong radial gradient. We interpret these trends using Milky Way analogs from the TNG50 cosmological simulation, identifying one that closely matches the kinematic evolution of our Galaxy. This halo's protogalaxy spins up into a relatively thin and misaligned high-$\alpha$ disk at early times, which is subsequently heated and torqued by a major gas-rich merger. The merger contributes a large amount of low-metallicity gas and angular momentum, from which the kinematically cold low-$\alpha$ stellar disk is subsequently born. This simulated history parallels several observed features of the Milky Way, particularly the decisive 'GSE' merger that likely occurred at $z \approx 2$. Our results provide an all-sky perspective on the emerging picture of our Galaxy's three-phase formation, impelled by the three physical mechanisms of spinup, merger, and cooldown.