Hi Jeremy Sparrell,

Please see the below steps to set up push notifications in salesforce on a particular action. There are two ways which I suggest you:-

Step 1:- Please go to the setup and type Custom Notification Type and select it. Please create a new custom notification record if not created.

Create a process builder which will fire on a specific object record creation which you will select in a process builder object option and see the below images for reference:-

Step 2:- After creating a custom notification record you can also write a trigger on that particular object instead of process builder:-

*** Trigger ***

trigger PushNotificationTriggerOnCaseCreation on Case (after insert) {
    if((trigger.IsAfter)&&(trigger.IsInsert)){
        PushNotification.sendPushNotification();
    }
}

******** Apex Code *********

public class PushNotification {
    public static void sendPushNotification(){
        try{
            List<Case> caseRecord=new List<Case>();
            caseRecord = [select Id from Case where Id=:Trigger.New limit 1];

            system.debug('-------In sendCustomnotification Method------');
            Messaging.CustomNotification notificationObj = new Messaging.CustomNotification();
            notificationObj.setBody('New Case is created with id: '+ caseRecord[0].Id);
            notificationObj.setTitle('Case Creation');
            notificationObj.setSenderId(Userinfo.getUserId());
            CustomNotificationType type = [SELECT Id FROM CustomNotificationType WHERE DeveloperName = 'Custom_Notifatication'];
            notificationObj.setNotificationTypeId(type.id);
            notificationObj.setTargetId(caseRecord[0].Id);
            notificationObj.send(new Set<String> { Userinfo.getUserId() });
        }
        catch(Exception ex)
        {
            System.debug('Exception at line >>>> '+ex.getLineNumber()+'Exception >>>>>> '+ex.getMessage());
        }
    }
}

The output will be like:-

I hope you find the above solution helpful. If it does, please mark it as Best Answer to help others too.

Thanks and Regards,
Suraj Tripathi