My wife and I are planning a 13 night stay in Greece from September 25, 2018 - October 9, 2018.
We are not "beach" people but love, history, good wine, fine dining, romantic adventures and special cultural experiences. We normally stay at HomeAway or AirB&B rentals. We travel medium to upper class.
We have traveled extensively in Europe and North America.
Please advise me on the following itinerary with good point and possible problems.

Day 1 - Fly from US to Athens
Day 2 - Arrive Athens mid morning
Take early afternoon flight on Aegean Air to Santorini
Rent car at airport
Lodging rental seaside condo in Oia Greece
Day 3 - Santorini
Day 4 - Santorini
Day 5 - Santorini
Day 6 - Drop off Rental Car
Late morning high-speed ferry to Naxos
Pick up rental car
Lodging rental condo Naxos (WHAT ARE TO STAY)
Day 7 - Naxos
Day 8 - Naxos
Day 9 - Return rental car Naxos
Take morning flight Aegean Air Naxos to Athens
Rental car Athens airport
Drive to Nafplion , Greece
Lodging rental or luxury B&B
Day 10 -Lodging - Nafplion
Day 11- Return rental car to ATH airport and taxi to Athens
Lodging rental condo or apartment Plaka area
Day 12- Athens
Day 13- Athens
Day 14- Athens
Day 15- Uber to airport
Mid-morning flight back to USA
Arrive home around 8:30 PM Eastern Daylight

Please give me any advice on where to stay , what to do , what not to do and how that I might fine tune my trip. We are a couple in our late sixties and unfortunately this will probably be our only trip to Greece.

Thanks to all

In general, looks to me like a great trip!

You might want to see if you can include your flight to Santorini in your primary ticket, as that would mean that the first carrier would be responsible for getting you to Santorini if there's any problem with timing. And sometimes, at least, you get the more generous transatlantic baggage limits for your local flight.

It's not an apartment, but for Naxos, you might consider the excellent Hotel Grotta.
http://www.hotelgrotta.gr

FWIW, I found the Michelin Green Guide and Rough Guide the best for each of these areas. And there are lots of great trip reports on this board to help you decide what to do. I hope to post my report, which covers each of these areas, in the next few weeks; you might watch for it.

Once on Santorini, you can easily wait a day or two before picking up a rental car. And now, please be prepared: I’m about to step on a soapbox: Coming from the U.S., you will almost certainly have jet leg, so I would strongly encourage you to take public transportation for your first night or two. Although many people are not aware of it, there is mounting evidence that driving with jet lag is just as dangerous -- to yourself and others -- as driving drunk, and <b>nothing</b> you can do will prevent the microsleeps (which you might not even notice) that are the apparent culprit. Seriously -- NOT a good idea, no matter your prior experience! Unfortunately, I know too many people who have died or been seriously injured in accidents to which jet lag was a contributing factor, and I know too many people who will spend the rest of their lives dealing with the knowledge that they were responsible for accidents that resulted in multiple deaths, including those of children, all because they chose to drive with jet lag. The issue has become a bit of cause célèbre for me….

Only one night in Napflion? You need another night at least to see the antiquites in the area.

HappyTrvlr raises a good point -- sorry I missed that!

I'm not sure what you would cut to make more time for Nafplio, or whether it would make more sense to skip it. I can say that I'm glad I had a bit more time in Naxos than you will have and also more time in Athens. If it helps with your planning, I had 3 nights in Nafplio, which I enjoyed very much -- but I also truly enjoyed Naxos and Athens. I must admit that I was a bit less thrilled with Santorini, although I enjoyed some of my experiences there very much. As always, it comes down to what YOU want to do with the time you have.

No bad choices, IMO, just difficult ones.

I'm a bit confused by your post. Have you already rented a condo in Santorini? Or are you looking for suggestions there? I'd be tempted to cut one day from Santorini and add it to Nafplio, which quickly became one of my favorite cities in Greece a few years ago. It would be a shame to visit for only one night.

Some hotel recommendations:

Naxos: Nissaki Beach Hotel on St. George Beach. It 's a lovely hotel within walking distance of both beach and town.

Nafplio: Hotel Amphytron ( spelling), a small boutique hotel across the street from the harbor. Try to reserve one of their junior suites, which has a stunning view from the terrace. The hotel is only a few blocks from town.

Athens: We always stay at AVA Hotel and Suites, in the heart of the Plaka. Ask for an exclusive suite, which has a dead-on view of the back of The Acropolis. Lovely rooms, great staff, nice breakfast, excellent central location for exploring Athens.

It was not very clear but actually night # 9 and Night #10 are both in Nafplio.
I have not made any reservations yet, just planning.
Since this will be our only time to visit Greece, I want diversity to give my wife and me the "GREECE" Experience . Food, Drink, Scenery and History

Thank you for the suggestions so far.

Santorini has the justly famous "Caldera View" but please do not think that this island is the "Greece Experience" in the fullest sense -- it has the 4 elements you name, but you don't meet many Greeks (except shopkeepers & Hotel owners/managers) ... even your waiters may be Bulgarians. The view is stunning, but I agree that you can enjoy it fully in 2-3 nights, and Nafplio and its area is worth fully 3 days/nights. This is not based on brief encounters but many extensive trips, that have included both destinations.