The casual and cool places in Mykonos are pretty easy, but they have to go to Caprice (the bar) because it is a MUST for the greeks during Sunset. The restaurants that are more "cool" are The Sea Satin Market - Caprice, has amazing fish and is in "little Venice", towards the Windmills. There is another great restaurant on Paranga beach called Ahinaioi, which is great for lunch and has amazing fish. Finally, Filippis has great traditional Greek cuisine, and is located in the center and is very well known in Mykonos.
In Athens, there are several great restaurants, most of which I can't name off hand but there are several in the Gazi and Thissio areas (very close to the center) that are both casual and cool. They should be walking distance from the center or an easy taxi or metro ride. The two in Kolonaki I really like are Oikei, that has great "traditional" greek foods and is moderately priced. It is on Ploutarchou 15 in Kolonaki... The other is in Psyri called Taki 13, which happens to be on Taki street, number 13 (Taki 13, Psyrri, center).
Hope this helps and it isn't too late,