Greek Isles Cruise (Virgin Voyages)

The Acropolis and the Acropolis Museum are in different places and require different tickets. You'll probably need 2 hours for the Acropolis and an hour for the museum. You could do the museum the first day, though it's closer to the changing of the guard than is the Acropolis.

You're in luck, your cruise ship will be the only ship in port for your day in Rhodes. What are your interests?

We spent 5 nights on Rhodes last May. The port is at the Old Town walled city. It has lots of history of the Knight of the Crusades.

My wife and I are both teachers, so both of us are big into the history and cultural aspects of the places we visit. We’ve been trying our best to research with islands we should explore and which ones we would be best to find a beach and relax. What was your favorite thing to do in Rhodes?

The two good things we found to do were the Valley of the Butterflies and walking the streets of the Old Town. We visited the Valley of the Butterflies with a rental car. It was hard to find a parking space in May, so I can't imagine what it will be like in July. Plan to go there as early as possible. It will be breath of cool air. I'd suggest going as early as possible. Probably use a tour or private guide to get you there. It's about a 30 minute drive from town and you'll spend about 2 hours hiking there. We were surprised that the flora was very similar to where we live in Southern Oregon (madrone, sycamore, pine and fir trees).

The Old Town has lots to offer. Rhodes has a long history of ancient Greek, Roman, Byzantine, Crusader, Ottoman, and Italian rule. There's very little evidence left of the Greeks and Romans. The Ottomans lost the island after losing WW1, and the Italians took over. The Italians lost the island after WW2, then it became part of Greece. Most of the Old Town's Crusader sites were reconstructed during the Italian rule as a summer home for Mussolini. You can visit Crusader sites, walk the walls and moats, get lost in the meandering streets beyond the touristed areas. The first 5 to 10 blocks inside the walls from the cruise ship are all souvenir shops and tourist restaurants. Hippocrates Square is kind of the center of activity. The Jewish Synagogue Museum was closed for renovation while we were there, but there is a monument for the deported Jews of 1944 at Martyrs Plaza.

Rhodes has unique Dodecanesian dishes. We got lost trying to find Romios Restaurant (not Romeos Restaurant) but found Ta Kardasia instead. It had great small dishes for lunch.

You may have sufficient time to visit Lindos and then get back to Rhodes town for a few hours. We took a taxi to Lindos which waited for us for the return trip. It was expensive but we found another couple to share the ride.
You can easily spend the whole day wandering around Rhodes town.
Here are some images of both places.
Lindos and Rhodes https://www.flickr.com/photos/stanbr54/albums/72157634754696058/with/9344972603/

...big into the history and cultural aspects....

In Athens, consider a visit to the Ancient Agora as it is very close to the Acroplis, usually not very crowded, and it is the location of one of the best preserved temples from that era. In Mykonos, try to take a short boat ride to Delos as it is sometimes called "the birthplace of the gods". Also in Mykonos, if you stumble upon a restaurant / bar named "RHAPSODY", do pop in. Great place to get food / drink overlooking the water.

My experience with Lindos last May, the parking was impossible (they had our car park on a pile of dirt and the Acropolis entrance line was over 30 minutes waiting line with no shade (there were 4 cruise ships in port that day). We gave up and left.

We are staying in the Old Town in October and want to visit Lindos.

We don't drive (so far) when travelling as it is the "wrong" side of the road for us.

We've hired a driver for the day to take us to Lindos and wait while we visit the Acropolis, have lunch and wander the town. He will then take us to another couple of historical sites that we are interested in.

The cost is €350, which is obviously not cheap but to us it is worth it to not have to worry about driving, buses, taxis and just have someone at our beck and call for a day.

islands we should explore and which ones we would be best to find a beach and relax.

With only one day in each location (except for Istanbul), there's so much to see and do that I'd recommend saving your relaxation for the sea days on the ship, and spend your shore time exploring and experiencing your surroundings. All of these places are very different. Obviously in Kusadasi you'll see Ephesus -- I'd recommend Delos for Mykonos and wandering the old town in Rhodes. We were not enthralled by Lindos but there is a beach there. Not sure what I'd do with one day in Santorini -- a shore excursion may be best here. But Chania is another good place to wander. A beach is walkable from the old town if you want to explore for a bit, and then have some beach time.

Kusadasi- Make sure that you see Ephesus while here. It is a "not to be missed site" Avoid any shore excursions or tours that combine it with "the house of Virgin Mary" as this is a huge time waster when you could be spending the time in Ephesus. The terrace houses are worth the extra ticket at Ephesus.

You are completely on the right track to do the Acropolis first thing in the morning. I went twice. Once in the afternoon when the paths up and down were solid masses of humans, and once right when it opened when you could actually enjoy the site. It is just a 5-7minute walk from the Acropolis entrance to the Acropolis Museum. The Museum is right next to metro entrance.

I went to the Changing of the Guard and then realized I would see nothing due to the crowds that were already there. I left. If this is important to you, go early and get in position. If it is just something you thought might be fun, skip it.