Agree to book directly with airline, would add that we always have seat selected when we complete our reservation. No "get them at the airport," did that once or twice and we wound up in different parts of the plane.
Although I ultimately book through the airline, I will typically do research ahead of time using Google flights to see the varying options you can get. You can use a search filter by airline and your origin city. You can get a price matrix, too.
Once you see the different ways you can get in and out of Greece, and the prices, you can then book directly with the airline (Google Flights has an option "book with (e.g.) United" and you are redirected to the airline. I have checked to make sure the click from Google is not adding a fare increase. When I have replicated the flights on the UAL website, it is the same prices.