The following table displays the permissions required by users for approval record types:

Record Type VisibilityEnabled
Approval_Process__c.ApprovalProcessTRUE
Approval_Process__c.EntryCriteriaTRUE
Approval_Process__c.FinalActionTRUE
Approval_Process__c.InitialSubmissionActionTRUE
Approval_Process__c.InstanceTRUE
Approval_Process__c.StepTRUE
Approval_Process__c.StepFilterCriteriaTRUE
Approval_Process__c.StepGroupTRUE
Approval_Process__c.StepGroupActionTRUE

For information on setting record type visibilities, refer to this Salesforce article.