How to pull from 'Person' field in one column to populate 'Department' of another

BShepard22 If you are not going to use those department values later, for filtering/sorting, etc. (you don't need to "save" department in column) and you just want to show the department in SharePoint list view then you can achieve it using JSON column formatting: 

{
   "$schema": "https://developer.microsoft.com/json-schemas/sp/v2/column-formatting.schema.json",
   "elmType": "div",
   "txtContent": "[$PersonGroupCol.department]"
}

Where [$PersonGroupCol] is an internal name of your person column in SharePoint list in this format: [$InternalNameOfColumn]. You can get the internal name of your SharePoint list columns by following this article: How to find the Internal name of columns in SharePoint Online?

If you want to "save" the actual value in department column, you can create a Microsoft power automate flow to update the department field based on the Person or Group field OR

You can customize the list form using Power Apps to auto-populate department column using Office 365 Users connector and Power Apps formulas. Check: Customize a Microsoft Lists or SharePoint form by using Power Apps 

Similar thread: Get Email from user and enter into SharePoint list field 

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