Over five thousand exoplanets have been discovered, but relatively little is known about their atmospheric properties and compositions. Characterizing exoplanet atmospheres is the most promising avenue through which we can learn more about exoplanets themselves, and analyzing high-resolution spectra through Doppler cross-correlation is currently one of the best methods by which this can be accomplished. In this thesis, I use state-of-the-art ground-based spectrographs in the optical and near-infrared wavelength regimes to detect, characterize, and understand the atmospheres of exoplanets. My focus is on super-Earths and hot Jupiters, for which there are no Solar System analogues and which remain poorly understood. I begin by characterizing the atmospheres of two warm, sub-Saturn mass exoplanets with high-resolution optical spectroscopy, pushing current observational techniques to lower masses and temperatures and exploring the effects of clouds on high-resolution optical spectra. I then investigate the atmospheric composition of a hot super-Earth with high-resolution near-infrared spectroscopy, providing stringent constraints on its chemical composition and presenting the first atmospheric characterization of an exoplanet from a new near-infrared spectrograph. I also investigate the efficacy of telluric removal methods across a broad, near-infrared wavelength range. Lastly, I turn my attention to the atmosphere of a highly irradiated ultra-hot Jupiter with high-resolution optical spectroscopy, resulting in a new detection of ionized calcium indicative of atmospheric outflows and/or a hotter-than-expected temperature. I then carry out a detailed search for atomic and molecular species in this planet's atmosphere, shedding light on its chemical composition and setting the stage for a multi-year optical survey of exoplanet atmospheres at high spectral resolution. These works expand our knowledge and understanding of exoplanet atmospheres, and shed light on the enigmatic populations of super-Earths and hot Jupiters. They also introduce novel techniques across a range of new instruments and surveys, paving the way for future studies in the field and opening the doors to an era of comparative exoplanetology.