1. A computer-implemented method, comprising: receiving, by one or more configured computing systems of a computing resource service, a prioritization policy from a customer of the computing resource service for use in scaling a group of computing nodes provided to the customer by the computing resource service, wherein each of a plurality of scaling policies specifies one or more associated changes to be performed to the group if criteria are satisfied for the scaling policy that include at least one resource utilization metric exceeding a threshold, and wherein the prioritization policy specifies priorities between the plurality of scaling policies; receiving, by the one or more configured computing systems, information indicating multiple alarms that have been triggered corresponding to multiple scaling policies of the plurality, based on the criteria for the multiple scaling policies being satisfied by monitored values for one or more resource utilization metrics; selecting, by the one or more configured computing systems and based at least in part on the specified priorities of the prioritization policy, a scaling policy from the multiple scaling policies corresponding to the triggered multiple alarms, wherein the selected scaling policy has a higher priority in the prioritization policy than one or more other unselected scaling policies of the multiple scaling policies; determining, by the one or more configured computing systems, a change to the group from the one or more associated changes for the selected scaling policy; and initiating, by the one or more configured computing systems and based on the higher priority of the selected scaling policy in the prioritization policy, the determined change in the group without initiating additional changes to the group that are associated with the one or more other unselected scaling policies of the multiple scaling policies.