Here are few links that you might find helpful :
A startup kit for new cruisers in RCL that @Matt created:
Live blogs for Greek Island https://www.royalcaribbeanblog.com/boards/index.php?/topic/9757-its-all-greek-to-merhapsody-november-3-10/ https://www.royalcaribbeanblog.com/boards/index.php?/topic/9042-rhapsody-of-the-seas-greek-islands-25-aug-1-sep-semi-live-blog/
Regarding each port you have:
Rome /Civitavecchia - In Rome you should plan to be at least two to three days so you can see the major attractions , One day for the Vatican , one day to tour Rome with the hop/on off buss , one day for the old area + basic attractions. As you can see even three days are very tight. take into consideration its very hot there in June and many palace do not have air conditions (even not in the public transportation). Civitavchia is around 1.5 hour drive from Rome , nothing to do there so its better to book a transportation from Rome to the port at early morning.
Santorini - The must do place is to go to Oia village , this is the one with the famous blue roofs and wonderful place for selfies 🙂 , its around 45 minutes drive from the port , tours from the ship starts at 43 Dollars.
Mykonos - Here its very simple , the tender boat will bring you to the village , everything is waking distance from the port , you can also go to one of the nice beaches around. if you just stay a the village you do not need any tour .
Athens - you will dock at Piraeus which is around 30 minutes from Athens - the must do here is the Acropolis , You can do it by yourself (you can take a ride from the port) but expect long lines to buy the tickets and go up the mountain , also you need to consider its a moderate walk up the temple.Here the ship offers tours that start around 75 $. There many other places to see near by . Another tip about Athens that its the best place to buy souvenirs and other stuff (Greek Olives and olive oil in the supermarket , cloth , etc) comparing to Athens or the other Island you will visit are more expensive.
Olympia - You can go and see the Olympia old city , was no there so I can not add any recommendations.
Regarding tours from the ship , you should visit the cruise planner site for your cruise every week or so , from time to time RCL will offer discounts on tour that can go up to 25% less .You can always book , and if you find it cheaper you can cancel and re book without extra fee (up to three days before the cruise).
Also take some time to rest , do not try to do it all , otherwise it can become a very stressful cruise with all the ports 🙂
Enjoy , its a very nice cruise.
@Traveler linked to my blog about my trip in November. When we were in Athens, we used GreekingMe for an independent tour, and I cannot recommend them highly enough! They have several tour options, depending on your interests. There is absolutely no shade on the Acropolis, so be sure to take water and a hat and wear sunscreen! The views of the city are amazing from up there, too.
I've been to Rome in June, and it is murderously hot then, and a perfect excuse to seek out gelato. Expect huge lines for St Peter's and the Vatican Museum unless you book something in advance. My perception of Rome may be a little skewed, though, as the two times I've been have been either with 4 kids or a pregnant wife.
We are going on a cruise departing from Civitavecchia this May, and am planning on going straight from the airport to the cruise port on arrival, and then doing a day of sightseeing in Rome on disembarkation day since our flight from FCO is late that night. I'll report back on that, but it may be too close to your cruise for you to make changes.
Your teens will have a great time. Make sure they go to the first night at the kids area (dont remember exactly how old they are, but they may be in different groups) ...they will meet lots of people that they will hang with the whole cruise. My daughter usually would meet her friends at the teen center every night and they would go in and out. Let me know if you have any specific questions...we did vision and freedom, so not the big ships yet, but alot is the same for teens! Jane