You have no time to waste in making arrangements, mainly because your trip runs in to High Season. ALso because your trip is short, logistics become extremely important -- and in Greece, logistics are more complicated than in other European countries -- for one thing, virtually no train service, and for another, ferries don't speed like cars, trains & busses. If you are not careful, transport could eat up days of your precious time.
Because not only days, but hours, affect your options, I'm not going to give specific suggstions until you share some vital details, i.e. what is the exact date (AND time of day) you land in Greece. and date/time of day your departure flight. (It really matters whether u land at 9 am or 4 pm). Where in US?? People from LA are MUCH more exhausted on arrival than people from NYC.
If as I fear, your 10 days includes flight, you may only have 7 or 8 days in Greece. That's enough for only one island plus Athens. If you want to dive into Greek life and embrace local culture, I advise u do not stay on Santorini ... it has a famous Caldera View but is totally touricized, and in high season jampacked & Hot 100°+. However, if you choose for your island Naxos or Paros, both these have 2 companies that do day-excursions to Santorini, so you could glimpse the view w.o. staying overnight.
Since u are starting so late, you need to make transport as efficient as possible, which means grabbing a domestic flight to your chosen island just as soon as you arrive at ATH airport. Perhaps flying back as well... main thing is to do Athens last, so you won't waste precious hours going from Airport-Central athens-back twice. One other possibility, if the timing makes it possible, would be to take a high-speed ferry to that Single island (such as Sifnos), and high-speed ferry back. Sifnos has no airport and unlike Mykonos & Santorini, harbors can't take giant cruise ships, so a bit more of that authenticity you value.
You need to get the sequence & transport solved before u book an Island hotel, so devote your first priority to that. You can book an Athens hotel for the last 3 nights without risk (because most hotels allow cancellations without penalty until a week or 2 before arrival). You'll want to find a hotel room very near (walking distance) to the main landmarks (Acropolis etc) for a couple of reasons (1) with limited time, you don't want to have to taxi or Metro for sightseeing and (2) it will be HOT, so it's good to be near your hotel when you need to collapse for a bit. To get the best advice, please share (in Euros) your preferred price-range per night ... otherwise people may recommend hotels too budgety, OR too luxe for your taste.
Everyone's lives are busy, but could I suggest you stop by a library and check out any Greece guides on the shelf??? Guides really ARE helpful as a filter ... the best ones are candid, and can help you sort out what's important to you. Too often, internet sites are designed to attract eyeballs, and don't mention important minusses as well as the plusses, of many destinations.