Abstract: Any realistic quantum system interacts with its environment. While this leads to inadvertent perturbations in many applications, this interaction lies at the heart of countless interesting phenomena. Trapped atomic ions are well suited to study quantum dynamics on a fundamental level. They feature extensive control in preparation, coherent manipulation, and detection of internal (electronic) and external (motional) degrees of freedom. In addition, they allow for a remarkable degree of isolation from their surroundings. In this thesis, we use a trapped-ion setup to experimentally compose and study specific open and closed quantum systems. Starting with a single magnesium ion, we introduce an incoherent interaction with a large environment to tailor an open quantum system. We demonstrate that the fast and substantial impact of decoherence can be exploited to devise a high-precision spectroscopy method with sensitivity to single photons. We measure one-, two-, and three-photon transitions in 25Mg+ with a relative frequency uncertainty below 5 x 10^(-9), making our results relevant for the interpretation of astrophysical data in the search of spatial and temporal variations of the fine structure constant. Further, in a bottom up approach, we use a hybrid ion crystal of magnesium isotopes to construct a nearly closed quantum system atom by atom. We choose a subsystem of interest and realize a tunable degree of isolation from the rest of the closed system to study properties of the (open) subsystem. By the deliberate disregard of correlations and coherences through the measurement of subsystem observables, we find evidence of equilibration and thermalization, despite the underlying unitary dynamics of the total system. We study their dependence on the engineered size of the environment and the controlled strength of the interaction, where the coherent evolution explores Hilbert space dimensions exceeding 10^6. Our time-resolved measurements demonstrate finite-size effects as well as the transition to mesoscopic system dynamics and highlight the importance of associated time scales. Our setup can be extended by addition of spins and bosonic degrees of freedom, preparation of arbitrary internal and external states and initial correlations, and full state tomography. This paves the way, both for the study of open and closed system dynamics and their properties, and for an analogue quantum simulator for various model systems