Appropriate clothing is essentially whatever is appropriate in warm weather clothing where you live. You need to be comfortable. Shorts are OK but should not be too short. If you visit a religious's site you need to have bare arms and legs covered. In June the temp can push the high 90's F. A hat would be a good idea.
Mykonos. There are 4 things to consider doing. 1) Wander Mykonos town. Its a warren of back alleys very picturesque. No need for a tour. Stop at a seaside taverna and have a drink and just take it all in.
2) High end shopping.
3) Have a beach day on one of the many beaches outside Mykonos town. Back to your question about what to wear. On Mykonos beaches wear whatever you want including nothing at all.
4) Ancient Delos. It was one of the most important Religious centers in the ancient world. The site is impressive and in my opinion a must see right along with Delphi. You do not have to do a tour. There are
ferries from the old port in town. Take a morning ferry 19 euros return and wander the site and return in the early afternoon which will give you sufficient time to do one of options 1,2,3
Day trip Delos and Mykonos http://www.flickr.com/photos/stanbr54/sets/72157637922138466/
Rhodes.
Our ship docked about 200 meters from one of the old town gates. The view over the city walls from the ship was very impressive. Again no need for a ships tour. Get out and wander the old town, get lost, visit the street of the knights and the Grand Master palace. Find a taverna and sit and soak up the atmosphere. A tour to Lindos or just take a taxi would also be a good choice. If it were me I'd just stay in the old town.
Lindos and Rhodes http://www.flickr.com/photos/stanbr54/sets/72157634754696058/
Santorini. You need to know that the ship uses tenders here. No dock. You tender to the old harbour at Fira the main town. There is a cable car that takes you up 1000 feet to the top. Spend your time in Fira just wander. Find a taverna with a caldera view and take it all in. Or from the old harbour there used to be a sailing ship that takes you to the lovely village at Oia. Buses from Fira also take you to Oia.
Now for the bad news. Depending on how many ships are in the only way to get back to the tender dock is to take the cable car back down to the port. It is not unusual for an hour or two wait to get on the cable car.
Santorini is a place where you might consider a ships tour. The tender goes to the ferry port where buses meet you to take you to Fira, Oia and if the tour planners have any knowledge of what is important Akroteri the Minoen town buried in the Santorini eruption. Think Pompeii but much smaller.
When the tour is over the buses should take you back to the Ferry port for your tender. Make sure the tour gets you back to the ferry port. That is the key reason why you might want to incur the expense of booking a ships tour.
Santorini http://www.flickr.com/photos/stanbr54/sets/72157632154800782/
You didn't ask, but in Athens, get to the Acropolis at opening 8 am I believe. Get your multi site ticket and get up to the site. You will have about an hour and a half before the tour buses arrive and the site gets crowded. Leave and walk down to the Plaka / Monestraki area. The rest of the sites on the multi site ticket are in this area. When you get tired find a taverna with an Acropolis view sit and take it all in.
A note on getting to the acropolis. There are several easy ways however to get there first thing in the morning from the Cruise ship port I would book a private transfer in advance. Taxi line ups are at the port but they are not at all interest in taking you to the Acropolis they want you to book a full day tour. We actually couldn't get a taxi to take us up. We had an excellent experience advance booking with Athens City Cab citycab.gr
Athens http://www.flickr.com/photos/stanbr54/sets/72157632121475515/
