I'm going to be backpacking Europe next year. I will be in Greece at the beginging of June. I have always been a huge fan of the greek culture and going there is going to be one of the highlites of my trip.
Where is some good places to go that I can see alot of Greece in my time frame(About 10 days). I'm not a huge tourist/musem person, I want to see the true culture, small towns, history, amazing old towns and structures.

Will I have time to go to the island and what are the costs of getting to and from the Island approx.?

I will be going to Athens for a few days for sure but other then that I don't know where to go.
I will be other flying or taking the ferry to Itlay afterwards.

Thanks for any help anyone can give me.