Collisionless shocks are ubiquitous in many space and astrophysical plasmas. However, since space shocks are largely steady-state, spacecraft are not well suited to studying shock formation in situ. This work is concerned with the generation and study in a laboratory setting of magnetized, quasi-perpendicular collisionless shocks relevant to space shocks. Experiments performed at the Large Plasma Device (LAPD) at UCLA and the Trident Laser Facility at Los Alamos National Laboratory (LANL) combined a magnetic piston driven by a high-energy laser (Raptor at UCLA or Trident at LANL) incident on a carbon target with a preformed, magnetized background plasma. Magnetic flux measurements and 2D hybrid simulations indicate that a magnetosonic pulse consistent with a low-Mach number collisionless shock was formed in the ambient plasma. The characteristics of the shock are analyzed and compared to other experiments in which no shock or a shock precursor formed. The results and simulations reveal that the various experimental conditions can be organized into weak and moderate coupling regimes, in which no shock forms, and a strong coupling regime, in which a full shock forms. A framework for studying these regimes and designing future shock experiments is devised. Early-time laser-plasma parameters necessary to characterize the different shock coupling regimes are studied through experiments performed at the LAPD and Phoenix laboratory at UCLA. In addition to spectroscopic and fast-gate filtered photography, the experiments utilize a custom Thomson scattering diagnostic, optimized for a novel electron density and temperature regime where the transition from collective to non-collective scattering could be spatially resolved. Data from the experiments and 3D analytic modeling indicate that the laser-plasma is best fit at early times with an isentropic, adiabatic fluid model and is consistent with a recombination-dominated plasma for which the electron temperature \(T_e \propto t^{-1}\). In addition, spectroscopic measurements and 1D radiation-MHD simulations reveal that the laser-plasma velocity distribution is segregated by charge state, with the fastest ion species having the highest charge state, but the slowest species having the most mass. The results also suggest that the fastest non-trivial ion population (the dominant charge state) at typical laser intensities (\(I_0 \sim 10^{11}\) W cm\(^{-2}\)) is CV (C\(^{+4}\)).