I just came from a Greek Isles tour from Norwegian Jade. As you are 6 it may make sense to rent a car (or rather 2) as you will be more independant and it will not be more expensive than using public transportation.
I would not even prebook, in all the mentioned ports it will be easy to get a car in every port. I was there 2 weeks ago - full high seasons - and plenty of cars were available.
Since we were only a party of 2 we sticked to public transportation:
-Corfu: Good idea to rent a car there as island is huge and nice beaches are remote and not easy reachable by public transport. If you want to visit Achileon palace, there would be a bus leaving from downtown Corfu, but not more than 1 per hour.
-Santorin: Ship docks in front of the old port and you tender to the old port. From there you need to get up to Fira by cable car (4 EUR), donkey (5 EUR) or by foot (NOT RECOMMANDED). In Fira you can take busses, taxis, rental cars. Usually you would go to Oia, busses leave there every 20 minutes and take roughly 30 minutes, however often full. Busses are comfortable and air conditioned. Beaches are so so, being a vulcanic island. Found Mykonos better for beaches. You may also take a boat tour from the old port, there are a lot of local boat companies directly in the port. However in our case we arrived at Santorin around 2pm and all "big tours" going to the volcano, hot springs etc were gone. Only tours that were offered were a speedboat to Oia with a transfer back to Fira for 20 EUR or so. If your ship arrives late as well and if you want to go to the volcano, you are stuck with the ships land excursions.
-Mykonos: Mykonos Town is huge and it's enjoyable to get lost there. Rental cars can't enter the town and need to park at the edge on parkings there. Nice beaches which are reachable easily by bus as well (however usually only every hour, when parties start in the late afternoon usually every 30 minutes in high season). If you want to go to Delos no need to rent a car.
-Katakolon: Nothing to do. You can rent a car and drive to Olympia (45 mins to one hour), or take a bus from a local travel agent (they are all around the place, 10 EUR return), or take a modern and comfortable train (10 EUR return as well). All take 45 minutes to 1 hour. We took the train and it was more than fine. No nice beaches there as far as I recall.
What impressed me the most was Santorin, especially Oia and the landscape which is dramatic. Corfu old town is strinking, too. Mykonos town, although being touristic, is very nice to stroll around. Mykonos has great beaches, too. Katakolon is pretty low key... We did all with public transportation which was generally cheap and very modern and easy to understand, especially in Santorin. However schedules are sometimes sparse (Santorin is quite ok however, especially between Fira and Oia).
In any case: I would recommend to make sure to leave ship early in the port, especially in Santorin as you will need to take the cable car and to tender, so make sure to get a early tender ticket to avoid huge lines at the cable car. In the other ports it makes sense as well, as if you miss your bus you will have loose one hour to get the next one. Same in Katakolon to get to olympia. Again, with such a big party a rental car may be a good option in my opinion. However since - at least with Norwegian Jade - stops in the ports are rather short, better don't loose too much time and get down the ship and into port early.
If there is anything else I can be helpful with, let me know