In this paper, we address the problem of bounding conditional expectations when moment information of the underlying distribution and the random event conditioned upon are given. To this end, we propose an adapted version of the generalized moment problem which deals with this conditional information through a simple transformation. By exploiting conic duality, we obtain sharp bounds that can be used for distribution-free decision-making under uncertainty. Additionally, we derive computationally tractable mathematical programs for distributionally robust optimization (DRO) with side information by leveraging core ideas from ambiguity-averse uncertainty quantification and robust optimization, establishing a moment-based DRO framework for prescriptive stochastic programming.