How to change the display of my field? I have a field in one of my content types called lab_test_id. When Editors add content of that type they enter all the ids of the lab tests associated with that node. I need to be able to take that id, call that lab test database, and display the lab test fields I get back.

Can I accomplish that by changing the display of the lab_test_id field via hook_field_display_alter or should I find another way to accomplish that?
I think you can do this using hook_field_display_alter, however, I would personally prefer to create a custom formatter for the used field type. With the main advantage that you can save settings for the field formatter.

See the answers of How do I write an additional field formatter for an existing field for basic steps on how to create your own field formatter.