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In this thesis, I present experiments on making and probing strongly correlated gases of ultracold atoms in an optical lattice with engineered potentials and dynamics. The quantum gas microscope first developed in our lab enables single-site resolution imaging and manipulation of atoms in a two-dimensional lattice, offering an ideal platform for quantum simulation of condensed matter systems. Here we demonstrate our abilities to generate optical potential with high precision and high resolution, and engineer coherent dynamics using photon assisted tunneling. We also create a system of bilayer quantum gases that brings new imaging capabilities and extends the possible range of our quantum simulation. To engineer precise optical potentials, we tackle uncontrolled disorder using in- coherent light sources and Fourier filtering of lattice beams. We develop a spatially incoherent light source which suppresses disorder caused by defects and scattering in the imaging system. Digital micro-mirror devices are used as spatial light modulators to shape arbitrary potentials with single site resolution. Next we study photon-assisted tunneling as an example of driven coherent dynamics. We observe sharp, interaction-shifted photon-assisted tunneling resonances, and resolve the multi-orbital shifts. Using photon-assisted tunneling, we drive a quantum phase transition between a paramagnet and an anti-ferromagnet, and observe quench dynamics at a critical point. We prepare tunnel-coupled bilayer systems, and use interaction blockade to engineer occupation-dependent inter-plane transport. The site resolved imaging of the bilayer system allows us to circumvent the limitations of parity imaging to directly observe the Mott insulator \wedding cake structure and density ordering in the anti-ferromagnetic state, and to perform spin-resolved readout of a hyperfine mixture.