One of my favorite places to go out in Athens for a drink (but also dinner or brunch) is the Saint Irene Square (Plateia Agias Eirinis).
The beautiful, old buildings dating from the 1900s have recently been renovated and now house bars, cafés, restaurants and clubs popular with (young) Athenians.
Agia Irini Square – Athens
Here are some places that I recommend …
Carnet de Dame Catherine 