Hurray for you! I did my first trip to Greece essentially solo (after decades of books/films/classes) and loved it so, I went back 12 times. You will have a Fab time.. a great place for women travelilng solo. Vets like us always recommend doing Island FIRST -- on arrival, allowing 2-hr window, take afternoon flilght to NAXOS, for at least 6 days/7 nights. Then, back to Athens (flilghts are 40 mins, big ferries are 5.5 to 6 hours... study & choose). Island time helps u acclimate, learn Greek ways, you'll be more savvy for naviagting athens & Day trips (omor on those later).
NAXOS ROOMS -- As Stanbr suggests booking.com is great for browsing -- rquest Naxos St George Beach, put in your approximate Naxos dates ( & when a list pops up, instead click "show on Map" -- that way you see blue "balloons" showing available places in Exact locations. BTW Galini has a nicebreakfast but only 1-2 rooms with sea v iew b alconies, and they are booked solid years ahead. Take a look at Hotel Palatia ... its upper floors have sea-view balconioes .... Aso avialable right on beach 1st week Sept is Thomais Studios and a real treat, Flisvos Beach Studios (latter are little rooms right behind the beach Sport Center -- it has terrace bar that's a v friendly sunset hangout. But don't dither! All are bargans & will go fast; book for free cancellation/no risk.
NAXOS ACTIVITIES -- Going Solo, I highly recommend Naxos TRansit's 1-day tour - http://www.naxosbustransfer.com/ - 9-5 it covers all island's amazing scenic/historic highlights w a noonday SWIM ... and you'll also meet other interesting independent travelers you can do other things with. As sequence I suggest for evening #1 a sunset walk to town/arch, day #2 explore port "maze" to castle/museum/cathedral etc +swims & day 3 - bus tour. Plaka beach is my fave outer beach -- Plaka sunset dining at edge of sea @ Niko & Maria's. the best. That's just a start.
ATHENS - I'd suggest doing yourAthens time the last 4 days of your visit ... when those world-famouos landmarks, and the museum treasures, will resonate with you most... and you can just get a super-early x95 bus or a flat-fee car service back to airport in less than an hour for your homebound flight. For lodgings - again recommend browsing booking.com for Oct 11-15 ... I tend to favor hotels SOUTH of Acropolis -- handy to everything but away from busy traffic, more greenery, lovely long promenade. And booking.com now has besides hotels, many Private rooms/apts available in that area w balcony views & great room prices. And with the safeguard of booking.com rules about cancellations etc. For moderate hotels around there, check Hotel Hera, Airotel-Parthenon, hotels Phillipos & Herodion. If my darlilng budget Hotel Phaedra has magical room 323, grab it & tell them Janet sent you. .. besides room balcony w Acrop view, has amazing rooftop canopied terrace open 24/7... great to bring snacks & a book.
BEYOND-ATHENS -- The BEST trip for a classics buff is 3 days in NAFPLION -- Gorgeous -- considered THE most beautiful Old Town in All Greece ... at tip of peninsula, susrrounded by sea, in the heart of Greece's Most Historic Ancient sites (Mycenae, Turins, Epidaurus, Nemea, etc )... and yet its only 2 1/2 hrs away by modern reserved-seat intercity bus! This amazing noncommerical website -- http://www.visitnafplio.com/ -- tells you all you need to know, how to get there, what to do, what to see, where to eat and more! Website & I agree, top hotel choice (and not superpriceY) is Agamemnon right on seafront -- another great choice is Omorfo Poli. I suggest coming from Naxos, going straight tto KTEL Kifissou Bus Terminal from Airport... and having 3-4 glorious days.
That's all I can include now, but if this plan appeals, come back for more...