Take bus X95 (3.20 euro) from the airport direct to Syntagma square. Your hotel must be in walking distance from there anyway if you want to have the basic districts and monuments around. But even if you live somewhere else there will be for sure at least one bus leading to the center. The traffic is huge some days though so it’s better not to loose all your day in the bus, especially if you don’t have many days to spend in Athens,
I will make a small itinenary for you but you have to read my tips for detailed description.
Day 1:At Syntagma square see the Parliament and the Tomb of the Uknown soldier, walk into National Garden, see Zapeion Hall and the Arch of Hadrian. Go down Dionysiou Areopagitou pedestrian street or from Syntagma Square go down Ermou Street for shopping. In both options visit Monastiraki area for the market, Psiri (for the cafes), Plaka(for the buildings, and Acropolis at the top of the hill, the Tower of The Winds in the Roman Agora etc) .
Day 2:From Syntagma square in walking distance also is Kolonaki district, the expensive central district, you can go there through Vas.Sofias avenue and if your wallet cant afford the prices (even coffee is more expensive here) go up to Lycabettus Hill for an amazing view of the city or walk down to Eksarxeia district (for the cafes, pubs, rock clubs etc)
Day 3:Walk in the center, from Sintagma square to Omonoia square through El.Venizelos Avenue and from Omonoia to Monastiraki through Athinas Street where you can check the meat and fish markets.
Time to visit some museums. Archeological, Byzantinum, War Museum and a dozen of other museums and galleries in many places basically in the center.
Day 4:Visit Sounio by bus for the Temple of Posidon
Day 5:Take an organised tour to
Ancient Corinth or to Delphi.
If you have a car visit Sanctuary of Amphiarion at
Kalamos and eat fresh fish at
Skala Oropou,
Dhilessi or
Paralia Avlidhos. Dont forget to swim at one of the beach along the way. If you have time take the ferry from Skala Oropou to
Eretria for more sightseeing and return to Athens from
Chalkida.
If you only have 2 days try to combine the first 3 days shown above. It's easy to do but not so relaxing.