For Mykonos you can rent anything from a scooter to an ATV to a car, depending on what you want to pay. The beaches can easily be reached riding any one of these things.

I would recommend Paradise Beach or Super Paradise Beach for the day. Both are lots of either fun or just relaxing, whatever you prefer.

As Aquamarine said, definitely see Ephesus when you stop at Kusadasi! It will be the best ruins you will ever see and then you can skip most of the ruins in Greece. The house where the Virgin Mary lived is near there too.

Be prepared for the heat in August! We are in the Greek islands now and it has been over 100 degrees some days in July.

Enjoy your cruise!