Variational inference has become an increasingly attractive, computationally efficient alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of accuracy measures for variational approximations that are both theoretically justified and computationally efficient. In this paper, we provide rigorous bounds on the error of posterior mean and uncertainty estimates that arise from full-distribution approximations, as in variational inference. Our bounds are widely applicable as they require only that the approximating and exact posteriors have polynomial moments. Our bounds are computationally efficient for variational inference in that they require only standard values from variational objectives, straightforward analytic calculations, and simple Monte Carlo estimates. We show that our analysis naturally leads to a new and improved workflow for variational inference. Finally, we demonstrate the utility of our proposed workflow and error bounds on a real-data example with a widely used multilevel hierarchical model.