stay as close as possible to the centre near SYntagma Square area (constitution square in English)so that you can easily walk to the Plaka area which by day or by night is very pleasant(old village-like area on slopes of Acropolis,buzzing by night) Also the Hill of Lycabettus -worth a couple of hours walk/funicular.
The Parthenon/Acropolis is easily walked from centre.
Avoid grotty Ommonia area with its cheapy hotels.
Good trips from Athens
- full day three island tour taking in isles of Aegina, Hydra, pROS
- Full day trip Epidaurus, Corinth,Mycenae fascinatiiong for history buffs -rather long and tiring for others)
- full day trip to delphi - long drive ,nice scenery, great scenic location.
-half day trip along coast to Cape Sounion
very good metro sytem in Athens - worth popping down to the little ports around in pireus area -or to northern suburb of Kifissia for evening tavernas with no tourists around.
Much much to do - if you read up before hand. Unprepared and ill-read visitors are put off by noise,traffic and heat. I love it....
...but stay in the centre around Syntagma - not far out or in Ommonia.
Great weather in September.