I was wondering if there was a tutorial on how to implement a REST API? I have been asked to set this up for our building department to use for a permit system and until recently have never heard of this. Any information would help.
It seems like your question is about creating/implementing a REST API in general and not something specific to an Esri REST API. If so, I suggest doing a quick web search on designing and creating a REST API, there are plenty of good primers, tutorials, and discussions. Esri doesn't put out any documentation or training on creating a generic REST API.
This may be a good place to start: https://developers.arcgis.com/rest/services-reference/get-started-with-the-services-directory.htm
To determine the organization ID for your ArcGIS Online organization, go to the Item Details page of any hosted item, and then locate the URL of the item on the bottom right-hand corner of the page. This URL will contain the unique ID for your ArcGIS Online organization, and from here, you can access the entire REST directory assuming you have admin credentials.
Hope this puts you on the right track!