Driven by ambitious renewable portfolio standards, variable energy resources (such as wind and solar) are expected to impose unprecedented levels of uncertainty to power system operations. The current practice of planning operations with deterministic optimization tools may be ill-suited for a future where uncertainty is abundant. To overcome the reliability challenges associated with the large-scale inclusion of renewable resources, we present a stochastic hierarchical planning (SHP) framework. This framework captures operations at day-ahead, short-term and hour-ahead timescales, as well as the interactions between decisions and stochastic processes across these timescales. While stochastic counterparts of individual optimization problems (e.g., unit commitment, economic dispatch) have been studied previously, this paper provides a comprehensive computational treatment of planning frameworks that are stitched together in a hierarchical setting, which parallels the widely accepted deterministic hierarchy of models. Computational experiments conducted with the NREL118 dataset reveal that, relative to its deterministic counterpart, the SHP framework significantly reduces unmet demand, and can lead to substantial savings in costs and greenhouse gas emissions. The latter improvements can be attributed to increased reliability of operations with lower reserve requirements.