Hi, I am actually going to give you my 3 favorite Greek Islands, not on the normal tourist track; Syros in the Cycladic Islands, and Patmos and Symi in the Dodecanese Islands. I suggest flying on domestic air to Syros. There are not flights between the islands. Every flight to a Greek Island, requires a flight to the island and then back to Athens with another flight to the next island. Luckily, now all flights are in one big terminal in Greece, and so it is much easier to pop back to Athens, never leave the airport and catch the next domestic flight out to the next island. You may be able to get a ferry from Patmos to Symi. I would go from north to south, i.e. from Patmos to Symi if you take a ferry, so you are not on board a ferry possibly bucking the predominant winds coming in from the north. Syros was a very wealthy island in the 1800's due to the many Sea Captains from this island that returned to the island building many buildings in Ermoupoli around the main harbor. Up in the hills, hidden from pirates, is Ano Syros, with the Cycladic white washed buildings, and little winding walking streets which are great to wander. At night, Bouzouki music can be found in the little bars in Ano Syros. Patmos, once part of the Northern Cycladic Islands, and now part of the Dodecanese Islands still sports the Cycladic Island architecture of cubist white buildings with cobalt blue trim. An important island in Christian history, this is the location where supposedly John the Apostle wrote the Book of Revelations in a cave, now an important religious shrine. On the top of the island is the Monastery of St. John, surrounded by the Old Town, another great location for wandering the walking streets. Patmos has a very local feel, with an active town around the harbor. Symi, owned by Venice for centuries, has a very Italianate feel, particularly in the architecture, with buildings painted in Tuscan colors. This little island has a great harbor town. Many herbs grow wild in Symi, which can be seen and smelled on a one of the great hikes available across the island, perhaps out to the Panormitis Monastery. Each of these islands has a lovely unique feel, are friendly, and well worth visiting for local island life.
There is one Greek island I really appreciate: Crete. I enjoy a lot the major Minoan archaeological sites and museums (one of the most fascinating and attractive period in the history of the Mediterranean Sea), but I also loved walking around its gorgeous landscapes.
The East part of the island is quite wild, with very moving places as Zakros and the ruins of its Minoan palace.
More to the West, Rethymno is a larger town, with impressive Medieval remains. Chania is an attractive place as well. If you are into adventurous expeditions, do not miss the most beautiful Samariá Gorge.
Concerning the other Aegean pearls, I recently fell upon this interesting article, describing lots of the Dodecanese Islands: http://www.leavingcairo.com/category/adventure-travel/dodecanese-islands-greece/ Lots of interesting tips about how to visit them and beautiful pictures!
I just returned this summer from our Greek odyssey. And I'm going to give you an alternative view of the major tourist islands, Mykonos and Santorini. There is a reason they are so popular, and if you plan right, you can avoid the crowds.
Santorini is one of the Mediterranean's most dramatic islands: a flooded caldera (a collapsed volcanic crater) with a long, steep, multicolored arc of cliffs, thrusting up a thousand feet above sea level. Sometimes called "The Devil's Isle," this unique place has captured visitors' imaginations for millennia and might have inspired the tales of Atlantis. But the otherworldly appeal of Santorini doesn't end with its setting. Perched along the ridgeline is a gaggle of perfectly placed whitewashed villages, punctuated with azure domes, that make this, undeniably, one of Greece's most scenic spots. If this place didn't exist, some brilliant fantasy painter would have to conjure it up. As soon as you get there, escape the main town Thira and head across the island toward the sunset. Best place to stay overnight, and watch the sun settle in the sea is Oia
Mikonosis the very picture of the perfect Greek island town: a humble seafront village crouched behind a sandy harbor, thickly layered with blinding-white stucco, bright-blue trim, and bursting-purple bougainvilleas. Mykonos' status as a fashionable, jet-set destination and a mecca for gay holiday-makers also gives it a certain hip cachet (and sky-high prices) — weary fisherfolk and tacky trinket stalls share the lanes with top-end fashion boutiques. The core of town is literally a maze, designed by the Mykonians centuries ago to discourage would-be invaders from finding their way. If you manage to break free, wander up to the windmills for the view, or head out to one of the many enticing sandy beaches around the island. Nearby is the island of Delos — one of the Greek islands' top ancient sites.
Hello from Rhodes. I would suggest you to go to Cyclades. There are many islands around there and very close to each other. You will start by ship from Pireas, is the big port of Athens, you can choose an island like Serifos, you can stay there few days and then you can go to many other island by ship in about 1-3 hours.