Ok, you are new to FileMaker, but how new are you to database design and/or programming/designing in other computer venues?
You can set up 2 or possibly three tables with relationships. Keeping it to two tables for now:
Employees>-----Departments ( ---< means one to many )
Employee::DepartmentID = Departments::DepartmentID (Departments::DepartmentID should be an auto-entered serial number)
You can define a supervisor email field in departments and your Send Mail dialog (You'd use this in a script) can then simply refer to Departments::SupervisorEmail for the To box in this part of the email set up.
It's also possible to simply put all your supervisors in the Employees label, add a 2nd table occurrence of Employees, named Supervisors which I'd link to a Departments::SupervisorID field, but let's stick with a simpler structure to get started.
If you have not already done so. Please look up Send Mail in FileMaker help and make sure that you can select Send Mail from the File Menu and use it to send an email to yourself. This test will confirm that you have a compatible Email account that you can use with FileMaker and that it is configured correctly.
After you get these initial steps working to your satisfaction, you may want to make a new post with your follow on questions where you spell out the structure of your data base in more detail. You can certainly add such a capability, but the exact details depend on the design of your database.