If you want to get from Turkey to Greece cheaply, you could ride a bus or take a train. But if being cramped in a seat for hours on end just isn't your style, you might want to consider taking a ferry to the Greek islands.
If you're in Turkey already, Bodrum, Cesme(near Izmir) and Marmara are the only cities with regular ferry routes to the islands. Decide which island you want to go to first. The Marmara lines sail to Rhodes, the Bodrum lines to Kos, and the Cesme lines to Chios. Going to Kos will give you the most options, as you can easily sail to some other islands from there.
Make your way to the Turkish city you'll be sailing from.
You can either go to the city port to book your trip or do so on the Internet. Greek Travel Pages (www.gtp.gr) has a helpful search feature in its ferry section that connects to the schedules of several different ferry companies.
For longer ferry rides, you will have the choice between getting a cabin and sleeping on deck. The decks usually have lounges with seats and are far cheaper than cabins -- though also far less comfortable. You will most certainly need to bring a reasonably warm jacket for being on deck; a sleeping bag is ideal if you're staying there overnight.
If you go to Kos, you will easily be able to get to Athens' port of Piraeus within a day on a ferry that will stop in Kalymnos and Astypalea along the way.
