The recent development of fermionic quantum gas microscopes has enabled studies of cold atom Fermi-Hubbard systems with single-site resolution, revealing a variety of interesting phenomena in regimes which are difficult to access with existing theory techniques. The Fermi-Hubbard model is of great intrinsic interest as a toy model for strongly correlated quantum physics, and may also describe the phenomenology of high-temperature superconducting materials such as the cuprates. Most experimental studies of cold atom Fermi-Hubbard systems have focused on probing equal-time spin and density correlations, but a wide region of the low temperature phase diagram may be better understood by exploring dynamical (unequal-time) properties. In this thesis, we first report on an experiment exploring the response of antiferromagnetic spin correlations to a magnetic field, and find evidence for short-range canted antiferromagnetic spin correlations. Then we turn our focus to probing response functions associated with unequal-time correlations relevant for understanding the pseudogap and strange metal regimes of Fermi-Hubbard systems. First, we describe the development of a technique to measure microscopic diffusion, and hence resistivity, in doped Mott insulators. We find that this resistivity exhibits a linear dependence on temperature and violates the Mott-Ioffe-Regel limit, two signatures of strange metallic behavior. Next, we report on the development of angle-resolved photoemission spectroscopy (ARPES) compatible with quantum gas microscopy and its application to studying pseudogap physics in an attractive Fermi-Hubbard system across the BEC-BCS crossover, setting the stage for future studies of the pseudogap regime in repulsive Hubbard systems.