If you don't book excursions through the ship, then you are on your own. That's how it works in the Caribbean cruises, so I assume it's the same for Europe cruises.
Actually in Europe the cruise lines offer shuttle service in several ports. Some cruise line may have a nominal fee, and others may be complimentary. Mykonos will probably have a shuttle. Santorini might be a tender port. Where it drops off there is a cable car to go to the city.