Mykonos is a good beach island and can easily be done on your own. This island is quite famous for its beaches, although be forewarned that its possible you might see some females going topless (its not real common, but its quite legal). One of our favorite beaches is called Paradise Beach and you can get there quite cheaply by taking the local bus. Since you have 5, it might be worth a few extra Euros to just take a taxi. As to the bus, you will need to go into town (about 1/2 mile from the ship) and than walk over to the bus station that is near the windmills (there is another station on the side of the town closer to the ship, but its the wrong bus station for Paradise. Paradise Beach is clean, sandy, and has lots of facilities including chair/umbrella rentals. There is even a swimming pool, although the Ocean water is so clean and delightful there is no reason to use the pool.
There are also quite a few excellent beaches on the island of Corfu. Our favorite here are the beaches near Paleokastritsa and it is about 4 miles from Corfu Town (the port).
As to touring on your own, its possible at any of your ports depending on your spirit of adventure. But, since you are a group of 5, you might want to arrange a car/driver/guide for some of these ports such as Athens. Its best to look around on the internet and make arrangements for tours, but I am worthless for this function since we do everything on our own and hardly ever take a tour (unusual for an American).