Most of the baryons, or "normal" matter, found in galaxies and galaxy clusters are found in the hot, X-ray emitting gas known as the circumgalactic medium (CGM) or intracluster medium (ICM). The hot gas traces the gravitational potential well and is affected by both thermal and gravitational processes, so we use observations of the hot gas to explore changes across the galaxy or cluster's radius. Heating and cooling in the central regions of galaxies and clusters is primarily driven by feedback processes, including Active Galactic Nuclei (AGNs) and Type Ia supernovae. We can use X-ray observations of the hot gas to understand its thermal history and how the various feedback mechanisms affect the gas at small and large radii. Furthermore, we use X-ray gas properties (temperature, density, entropy, concentration, centroid shift, and power ratios) to characterize galaxies and clusters, understand their evolution, and classify them in meaningful ways. The combination of observations along with theoretical models and simulations explored in this thesis provides key insight into understanding how feedback processes affect the hot gas.I begin by presenting gas property results for a uniformly reduced sample of 348 galaxy clusters and show how those results can be used to characterize the sample and for further galaxy cluster science. I will then turn my focus to early-type galaxies for the remainder of this work. I examine a sample of 12 nearby early-type galaxies with powerful radio sources and find that IC 4296 exhibits unusually low central entropy as previously observed in NGC 4261. We also find some evidence that the minimum of the ratio between the cooling time and free-fall time, if it occurs at the galaxy center, may indicate the presence of a powerful radio source. Finally, I examine the galactic atmospheres of a sample of 49 early-type galaxies. I will show that the equilibrium pressure and density radial profiles for single- and multiphase galaxies agree with the Voit et al. (2020) theoretical model. I also find evidence for a correlation between the central velocity dispersion and entropy profile slope of the galaxies in the sample that agrees with the theoretical model.
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