Whilst I was there a few months ago I mostly wandered around Athens city to get a feel of non-touristy spots, walked to the sea coast, the stadium, used the metro to get to and then explore historical sites (the Acropolis especially worth a visit), chilled in local coffee shops for a drink/bite to eat in the clubbing district; oh, and found me a mouth-droppingly hot greek adonis for the night.
Wasn't there very long at all though. If/when I go again, I'll be sure to read/visit the time out magazine website for Athens re other local events/things to during my stay. I'd recommend you do that.