Virtualized base stations (vBS) can be implemented in diverse commodity platforms and are expected to bring unprecedented operational flexibility and cost efficiency to the next generation of cellular networks. However, their widespread adoption is hampered by their complex configuration options that affect in a non-traditional fashion both their performance and their power consumption. Following an in-depth experimental analysis in a bespoke testbed, we characterize the vBS power consumption profile and reveal previously unknown couplings between their various control knobs. Motivated by these findings, we develop a Bayesian learning framework for the orchestration of vBSs and design two novel algorithms: (i) BP-vRAN, which employs online learning to balance the vBS performance and energy consumption, and (ii) SBP-vRAN, which augments our optimization approach with safe controls that maximize performance while respecting hard power constraints. We show that our approaches are data-efficient, i.e., converge an order of magnitude faster than state-of-the-art Deep Reinforcement Learning methods, and achieve optimal performance. We demonstrate the efficacy of these solutions in an experimental prototype using real traffic traces.