We’re flying from Athens to Santorini for the day with no luggage. Do we have to go through Immigration and how long should we be at the airport prior to leaving?
This is a domestic flight, so there is no immigration control: once you've entered Greece, you're free to move about as you wish by car, train, plane boat or anything else, just like almost any other country. However, you will have to show appropriate ID to board the flight and, for many foreign visitors, that essentially means your passport.
Your airline's website will advise you how early to check in. It's normally one to two hours before a domestic flight.