Re: Athens
At the Acropolis, obviously, and the Acropolis museum close by. Close to there are the Agora (where the ancient Greeks did their shopping), Stoa of Attalus (Roman era long building with loads of Acropolitan remains), and the ancient Theatre of Dionysos.
And there are loads of other museums, e.g. the national archaeology museum (one of the world's greatest), Benaki (general Greek history to independence), Cycladic art for pre-classical island stuff. There's a brand-new modern art museum freshly opened too, literally this month. Even the metro stations sometimes have stuff - Syntagma Square's station has its own museum, Monastiriaki is on an archaeological dig.
Take a look at the royal palace, they ran out of money after building one floor. Tomb of the Unknown Soldier is in there. The place for pootling around is Plaka, all narrow streets and colourful houses to the north and east of the Acropolis.
If you had any spare days, I would recommend trying to get the catamaran to Hydra, an island an hour away from Piraeus harbour that is home to a jewel of a village. Still something of a secret.