If I understand you correctly.
All you would need to do I think is have an infopath form with the employee name etc,published to an SharePoint Form Library. On the form list the stages as individual fields. As per your post you could use a drop down box for each stage answer. The default value could be not started. Then you publish the form and drop down fields to your SP form library.
This way you could have a view with the employee name and each stage of the process as a column, in almost like a table and at a glance a manager or HR could see the new employee progress. Using a Save (Submit data connection) button, any changes made to the form are saved to the library.