Why not spend the entire trip in Turkey? More than enough to do there.

For flights inside Turkey, when I was there New Year's 2012-3 I found that the international Turkish Airlines site had wayyyyyy cheaper prices than I was finding on kayak, other sites et al. We're talking 29-79 TL for various flights. Just go to the international site instead of the US one which had the same flights for $200+. You will confirm your booking and then will need to call. I was able to pay and get it ticketed over the phone (my backup was that I live in LA and if need be, could have gotten to a Turkish Airlines office in person. Wasn't necessary though).

Another option if you are not tied to Athens but still want to see part of Greece is flying into Thessaloniki. Nice area and you can take a bus/train into Turkey cheap and easily. The Trakya (Thracian) region has some of the best music in the area, friendly people, and great food.

Info on buses/train connection should be available on seat61.com
My 2 lira...