ok, so having a wife from Athens and a home in Greece.
Go to Monastiraki and Plaka for a day (you have to), in the morning and then walk up to the Acropolis in the afternoon, having lunch on the way up. Monastiraki has a great leather market. This is the tourist thing to do, and you do need to watch your wallet, but it is essential, totally essential. you could even spend 2 days there, the museum and ruins at the foot of the Acropolis are astonishing.
Glyfada to the south is the posh seaside / beach residential area and could be worth an evening meal, and Kifissia and Kefalari in the northern suburbs are worth an afternoon shopping and perhaps a spot of lunch. use the metro to get around, it is good.
Day trip, personally Epidavros is fantastic, but I would also consider the Corinth Canal and Ancient Corinth, you can get a bus from Kifissos in Athens, or go on an organised tour.
Instead of Epidavros you may want to go to Mykines, the seat of king Agamemnon, with its lion gate, treasury and fantastic museum, and views over Argos, this is where Greek Mythology began. I would be surprised if a tour doesn't do Corinth and Mykines together. Corinth is 1 hour out of Athens, and Mykines a further 40 mins beyond that.
The villa of Herodotus Atticus is also a good day out.
Argos is not great, it is a dump, but Nafplio is amazing, just round the coast from Argos, an old medieval Venetian fortress town and the seat of the first Greek parliament in 1823.