Most island-bound travellers give Athens one night and consider it a chore. That is a mistake, and an easily fixed one.
Day one: the classical core, early
Be at the Acropolis for opening. In summer that means 8am, before the heat and the cruise groups. Ninety minutes is enough on the rock itself; then descend to the Acropolis Museum, which holds what the Parthenon lost and explains it well.
Afternoon: the Ancient Agora, quieter and more atmospheric than the Acropolis, with a beautifully preserved temple and a reconstructed stoa. Finish walking through Monastiraki towards Psyrri for dinner.
Day two: the modern city
Start at the National Archaeological Museum, which is the most important collection in the country and consistently less crowded than the Acropolis Museum. Two hours.
Then walk: Exarcheia for political murals and bookshops, Kolonaki for the cafés and the funicular up Lycabettus, Koukaki and Pangrati for the neighbourhoods where Athenians actually eat.
If you have a third day
Take the coastal tram or bus to Cape Sounion for the Temple of Poseidon at sunset, or the ferry to Aegina for an hour of island life without committing to the Cyclades.
For onward travel, see Greek island hopping by ferry.