Once you decide to discover Athens you should definetely experience a bicycle tour around the city. Begin your tour early in the morning and combining sightseeing and exercise.
Begin the tour from Dionysiou Areopagitou street towards the Thiseio and Kerameikos. Take the bike route from there to Thessalonikis street, head to SNFCC, the Tae Kwon Do arena, the Naval Museum (battleship Averof) and the Flisvos Marina. From ther you can continue along the coastline and getting back through Petralona and Plaka district.
Get my suggested walking tour map:
pay extra attention (no matter you take the bike route)
begin on morning hours
take pic – nic supplies and take a rest at SNFCC