The cerebellum regulates nonmotor behavior, but the routes of influence are not well characterized. Here we report a necessary role for the posterior cerebellum in guiding a reversal learning task through a network of diencephalic and neocortical structures, and in flexibility of free behavior. After chemogenetic inhibition of lobule VI vermis or hemispheric crus I Purkinje cells, mice could learn a water Y-maze but were impaired in ability to reverse their initial choice. To map targets of perturbation, we imaged c-Fos activation in cleared whole brains using light-sheet microscopy. Reversal learning activated diencephalic and associative neocortical regions. Distinctive subsets of structures were altered by perturbation of lobule VI (including thalamus and habenula) and crus I (including hypothalamus and prelimbic/orbital cortex), and both perturbations influenced anterior cingulate and infralimbic cortex. To identify functional networks, we used correlated variation in c-Fos activation within each group. Lobule VI inactivation weakened within-thalamus correlations, while crus I inactivation divided neocortical activity into sensorimotor and associative subnetworks. In both groups, high-throughput automated analysis of whole-body movement revealed deficiencies in across-day behavioral habituation to an open-field environment. Taken together, these experiments reveal brainwide systems for cerebellar influence that affect multiple flexible responses.