We measure empirical relationships between the local star formation rate (SFR) and properties of the star-forming molecular gas on 1.5kpc scales across 80 nearby galaxies. These relationships, commonly referred to as "star formation laws," aim at predicting the local SFR surface density from various combinations of molecular gas surface density, galactic orbital time, molecular cloud free fall time, and the interstellar medium dynamical equilibrium pressure. Leveraging a multiwavelength database built for the Physics at High Angular Resolution in Nearby Galaxies (PHANGS) survey, we measure these quantities consistently across all galaxies and quantify systematic uncertainties stemming from choices of SFR calibrations and the CO-to-H2 conversion factors. The star formation laws we examine show 0.3-0.4dex of intrinsic scatter, among which the molecular Kennicutt-Schmidt relation shows a ~10% larger scatter than the other three. The slope of this relation ranges {beta}~0.9-1.2, implying that the molecular gas depletion time remains roughly constant across the environments probed in our sample. The other relations have shallower slopes ({beta}~0.6-1.0), suggesting that the star formation efficiency per orbital time, the star formation efficiency per free fall time, and the pressure-to-SFR surface density ratio (i.e., the feedback yield) vary systematically with local molecular gas and SFR surface densities. Last but not least, the shapes of the star formation laws depend sensitively on methodological choices. Different choices of SFR calibrations can introduce systematic uncertainties of at least 10%-15% in the star formation law slopes and 0.15-0.25dex in their normalization, while the CO-to-H2 conversion factors can additionally produce uncertainties of 20%-25% for the slope and 0.10-0.20dex for the normalization.