Hi,
I'm so jealous - you're going to have a great time and September is a lovely time to see Athens, not too hot and humid.
I've never stayed there but Glyfada to the south of Athens looks nice, I've passed it on the bus and I think the metro goes there now.
Also, would you consider a day or two on Aegina or Angistri (I can recommend a hotel here if you're interested)? They are considered part of the commuter belt of Athens. You get the metro to Pireaus and hop on a catamaran and you'll be there in no time (< 1hr if I remember rightly). Worth the trip for the pistachios
Agia Marina on Aegina has a nice beach and some ferries go directly there without stopping in Aegina town.
As for where to stay, hotels (and self-catering) are dead cheap in September. It's no harm to book in advance, at least for your first night. And haggling will usually get you a better deal.
Are you planning to stay in the city centre at all? I can recommend a rough and ready hotel with a roof bar to die for. If you're happy with basic but clean rooms check out [broken link removed]. No affiliation apart from staying there every time I'm in Athens. 2 minutes walk from Monastiraki metro so very handy.
Matt's website gives a handy introduction to Athens and Greece in general.
Hope this helps,
g