Good for you TOni! I too wanted to go to greece for many decades, finally made it in my 60s, loved it so much have managed to come back 12 more times (!). the first few days of that first trip, a friend was along, but then 3 weeks on my own, and I managed just fine! Since then my visits have been mix, some with newcomers along, others on my own, but then meeting up with friends I'd made in the past. It all works... if you're sociable, you'll find other sociable travelers as well. And having lived in 3 of the biggest US cities of all, I can say I've never felt safer than I have in Athens and in the islands, well, one feels free as a breeze. GO for it!
Stanbr's advice is a great one, and I advise you not to dither... make your domestic flight bookings IMMEDIATELY! Try to get your initial transatlantic flight (Im guess u are from US or Canada) to be nonstop from N AMErica. WHy? Because with only 10 days, every hour is precious! THose nonstops go mainly from Eastern airports, and are the ONLY routes that arrive at ATH Airport in the AM -- from 9 - 10:30 depending on Airline. The ones that connect in Europe don't reach ATH til after lunch (stometimes late with delays), and by the time u get to an island, the day's gone. With arrival by mid-morning u can get a Santorini flight by 1 or 2, and be gazing at the caldera view from your room or a cafe by late afternoon. Even more urgent is the flight from NAXOS to ATH, because these are small (modern) planes, 45 psgr, and not many flights per day. Book NOW; flights are changeable if u need to tweak the dates.
LODGINGS -- Do NOT book these until your flights are set! You can always find another place to stay, you cannot always conjure up an airplane if all flights are booked. Come back to this thread as soon as you can tell us flight times, and we'll give you a range of available, affordable places in all 3 destinations (It will help if you can also say how much per night is in your budget).
ACTIVITIES -- Also when you've got those bookings, give u a quickie briefing on best sightseeing tips, with links. From experience, I do advocate for solo travelers to take some kind of tour on 1st morning after arrival -- because everyone else is like you, new to the place, also other singles... you often find someone to lunch with, shop with etc. You don't have to book ahead, just know what your options are & book the night before.
OK -- get busy, get back to us FAST... and we'll help you design a dream time for your FIRST trip to Greece!