I am a total noob when it comes to writing these unit tests. I would greatly appreciate any guidance to help me get this figured out.
Your trigger is set to run after a case record is updated however your unit test is only inserting cases and updating contacts. Think about how you would test your trigger in the UI and then attempt to do that via code.
Create a case.
Insert the case.
Modify the survey date.
Update the case.
Remember to bulk test as well as use system asserts to validate the expected outcome.