I have spent more than a year of my life in both places and love both.
My suggestions for Sicily are Taormina and Agrigento.
As for Greece, Santorini is an excellent suggestion, especially the village of Oia (spent 3 weeks in a house rental on the cliff-face there last summer), but I would not miss the island of Milos for the incredible beaches.
If you need hotel recommendations for Milos or Oia let me know. Otherwise, it's been a while since I was in Sicily and I can't remember.
Sept./Oct definitely over the spring. I swam in Taormina in November on an 80 degree day when the locals were wearing winter coats. Water was cold but warmer than, say, Provincetown.
And no, they don't hate Americans at all there. A lot of people do hate American foreign policy. There's a difference.
Personally, I would head to the Greek Islands over Sicily, unless you plan on touring the mainland (i.e. Rome, Florence, Venice).