We visited Mykonos as part of a 12-day Greece, Turkey, Italy cruise in July on Celebrity Reflection. Before taking our trip, I researched quite a bit to determine whether to take cruise ship excursions or on-our-own excursions. I had a difficult time finding postings with such recommendations so I'm sharing the logic I used to make the decisions and the results of our choices.
Unlike Santorini, the island of Mykonos is flat; there's no having to journey up 800 feet to get on the island. We opted for a day on-on-own and were pleased with the choice. Although we had to tender, the boat dropped us right at the Mykonos town. Being in Mykonos is like being on set of Mamma Mia. Everywhere you turn, it's quite picturesque with views of the crystal clear Aegean Sea or flowering plants.
A few pointers: To find the taxi-stand, wander along the waterfront and on the right-hand side you'll come to an open square. This is where the taxi stand is. It's not well-marked. If you want to journey to another part of the island, you can easily get a taxi here. We didn't realize this so kept walking up towards the main road, only to be told that we had to backtrack back to the taxi stand. Although we'd initially wanted to visit one of the legendary beaches, we gave up and decided just to enjoy Mykonos. We spent hours getting lost in the windy streets, looking through shops, and eating yummy Greek food. I wanted to swim in the Aegean and there is a small public beach right along the main street. It was clean and cool and fulfilled my goal.
Had I to do it over again, we could've have explored a different part of the island had I planned a little more ahead. If you want to do this, I'd recommend getting a map of Mykonos before you go. The only map we got from the cruise ship had the main shopping district (ie, Diamonds International stores) so it wasn't very useful. I would've felt more confident about getting away from the ship had I had a better understanding of the layout and travel time between the port and varying cities.