Acropolis (includes the Parthenon, Erechtheion, Stoa of Eumenes and Acropolis Museum), Theatre or Dionysos, Theatre of Herodes Atticus, Temple of Olympian Zeus, Ancient and Roman Agoras, Fiolopappos Hill, Zappeio and National Gardens, Parliament Building.
Quick Tips:
I would suggest not to spend more than two days in Athens (maybe stay on a nearby island and take day trip or stay for a weekend instead). Apparently Athens requires time to appreciate.
In high season, book accommodation in advance to avoid ending up in the disgustingly dirty and scary Hotel Vergina (or Hotel V*gina as I would call it).
Don't spend long in the souvenir shop-filled and touristy Plaka (old town).
Athens is not big on animal welfare. You will find dehydrated and sore-covered homeless dogs sprawled out in every alleyway and shady area.
Despite all this, Athens IS worth visiting, if not just for the Acropolis.
Best Way To Get Around:
Bus number E95 leaves Plateia Syntagmatos every 25 minutes to the airport.
All the above attractions can be reached by foot from one another. The metro is clean and frequent for travel further out. For the Acropolis, take the metro to the 'Akropoli' stop on the red line.