Athens,
Head to Centre of town, say Syntagma.
Acropolis : There is a nice old building called the Parthenon on an old hill called the acropolis : you will have heard of it.
Slightly down the north side of the Acropolis is the flea market which is worth a visit also.
I would also recommend visiting the National Gardens ( near Syntagma Square )and the National Museum.

For me, though, the best part of athens is wandering from small square to small square at random, as this way you start to feel the pulse of the city.