Athens-Santorini-Naxos-Athens over 2 weeks- October Itinerary Please

1. Where do we start? (Stay in Athens after arrival vs. fly/ferry to next destination and do Athens at the end)

If your flight schedule to Athens allows for it, catch a connection to your first island right away.

Ideally, though it's not mandatory, you would start with Santorini (the furthest from Athens), then take a ferry from Santorini to Naxos, and finally head back to Athens to stay before your flight home.
This "Santorini-Naxos-Athens" triptych is actually the classic, tried-and-true itinerary used by many North American travelers before you on this forum.

2 . Athens x 4 nights and 5 nights on each island?

5 nights in Santorini is probably at least one too many; save those nights for Naxos, which easily deserves 6 nights.

3 . Assuming we rent a car on both islands?

We always rent a car on every island, regardless of whether it's touristy or not, small or large. It gives you a freedom of movement that local bus transportation simply can't match, and it saves a lot of time. That being said, renting a car in Santorini isn't strictly necessary if you're just sticking to the main tourist highlights.

On Naxos, I know many people here are content staying near Chora (Naxos Town), shopping in the old town, and only visiting nearby beaches. However, if you aren't the homebody type a car is absolutely essential if you want to explore the island efficiently (especially for hiking). The other advantage of having a car is that it opens up countless possibilities for places to stay without needing to worry about proximity to a bus stop.

Which brings us to question 4:

4 . Beautiful places to stay with relatively easy access to restaurants etc? Nightlife not important- would opt to spend our evenings with a
glass of wine and simply take in the view

That comes down to a few hundred places, from hotel rooms in every category to locally-run accommodations like studios, small and large apartments, houses, and even villas with private pools, etc.
This choice increases tenfold if you have a car. It's up to you to choose based on your tastes, budget, and preferred style of accommodation.

5 . Is October too late? Will sights/restaurants be open?

Things start to quiet down starting mid-October, but in a way, that's all the better for you. And of course, there will be tons of restaurants, bars, and tavernas open. Even without the tourists the islands are lived-in 365 days a year.

6 . Transportation between islands and back to Athens? Is a ferry fine given we have time?

Of course, taking the ferry is part of the Greek island experience. Visiting the Greek islands without taking a ferry is like visiting Las Vegas without going to a casino, and Santorini and Naxos are served by multiple ferries daily to and from Athens.

Curiously, my wife and I just did a trip to Greece in June for our 40th. We did Athens-Santorini-Crete-Corfu-Athens with a day trip out to the Peloponnese in the second stay in Athens. Timing was good as it was the end of shoulder season and the weather was good. Cannot speak to October, though I know things close and slow down. Have heard early October is better, and September better than that.

In general, tap water in Greece is not potable, you will need to buy drinking water. Greek plumbing does not accept toilet paper, there will be a waste can next to all toilets. We got water and groceries at AB Supermarkets in Santorini and Corfu. Staff was helpful and they were reasonably easy to navigate. Regarding rental cars, hopefully, you can drive stick. Roads on any of the islands are narrow in places, especially in town, and there are many hills and switchbacks. I spent a lot of time in 2nd and 3rd gear. We had no issue using Google Maps.

We only took a ferry from Santorini to Crete. It was fine. The boarding process was a bit chaotic, but the ride was smooth and relatively fast.

Santorini

DO NOT bring large bags, backpacks work best as the streets are rough and there are many stairs.

The majority of hotels/suites are in Thera (Fira), Imerovigli, and Oia. Thera is where the cruise ships come in so it is the most crowded. Parking can be an issue there. We stayed in Imerovigli which is near Thera. There, I would recommend the Phaos suites. They have on-site parking. Because they face the caldera, they get sunrise views as opposed to sunset, the rooms are nice and the price is better. The apartments with the sunset views do not have direct access to parking so just getting to them is a challenge.

Oia has probably the best sunset views being out on the end of the island, but a car there might be a bit of a hassle. It does not lack for good restaurants. I can recommend Candouni where we had a great lunch, and Ambrosia, where we had our Anniversary dinner.

Highly recommend the hike from Thera to Oia. Doing it in the morning provides a bit of shade if you are leaving Thera/Imerovigli. Wear shoes with good grip, there is a lot of loose rock. I wore trail runners, my wife brought her hiking boots. You can take a bus back. Highly recommend taking a sunset cruise. These usually start in the afternoon and stop in several locations for swimming, and include food and beverage. The archeologic site at Akrotiri is worth a visit. They are just starting the excavation. From there, it is a short drive on a marginally paved road to the Red Sand Beach. Melini's Tavern there is a very nice spot on the water with good food. Two of the family members are artists, so there is also a gallery. Wineries suggest reservations, mainly because they have tour buses coming in all the time. Friends recommended Santos winery for dinner, we did not get there.

Athens

Good shoes are a must. Obviously for places like the Acropolis and Agora, but there are a lot of cobblestone streets and stone stairways which have been worn smooth by many feet over the ages. Grippy soles are a must and if it rains, I would be very careful. We had issues just with the dust on them. The city itself can seem a bit gritty...we lived in NYC for years so this did not bother us, but we know people who would be put off. We both loved Athens.

In Athens we stayed at Hotel Evripedes for the first stint. We had a large balcony overlooking the Acropolis, which was quite nice. The room was fine, nothing special. The rooftop bar was nice. Second stint we stayed in the Estia Boutique Apartments. Nicer room, smaller balcony without the view. Onsite parking. I would recommend either. Both close enough to Acropolis, Agora, and Monastiraki to walk. If you can, get tickets for a show at the Odeon of Herodes.

Hope this helps.

Cheers,
John

Some thoughts.
Many experienced visitors to Greece suggest you get out to your first island on the day you arrive. Athens is very intense and hectic so going to a relaxing island first is a good idea.
Santorini and Naxos are good companion islands. Santorini has the dazzle with the caldera and Naxos will give you more traditional Greek experience.
When we combine Santorini with Naxos we do 3 nights in Santorini. That is enough time for us and thats all our finances can handle.

Use the extra 2 nights on Naxos.
Naxos is our go to island. It has a wonderful main town Chora with a very good beach (St George) just a 10 minute stroll to Chora.

October is the beginning of the end of the season. One year we were staying at a hotel at Plaka beach (about 7km from Naxos town,well served by regular bus service.) It was about the 4th of October and we were the last guests. Honestly the hotel literally closed and locked the door as we left and started their end of season staff party.
We relocated at Naxos town and it was still open with lots of people. We still needed to have reservations at tavernas for dinner though.
If you love a glass of wine at sunset Naxos is going to be a treat for you.

You certainly can rent a car on Naxos but the bus service around the island is good. The bus company does a full day tour which visits many of the island landmarks. A car rental give you lots of scope to explore the island but is not something you absolutely need to do.
Some images https://www.flickr.com/photos/stanbr54/albums/72177720334044112/
And lots of sunset drinks
https://www.flickr.com/photos/stanbr54/albums/72177720329571437/

In general, tap water in Greece isn't safe to drink.

That's not true, tap water is actually safe to drink in Greece. The only catch is that it can taste pretty bad on the islands because of the treatment process.

Renting a car gives you a lot of flexibility to explore the island, but it isn't an absolute must.

Sorry, but that's another point I disagree with. It all comes down to what you mean by "explore." The way I look at it, no bus tour will let you efficiently explore Naxos, its beaches, villages, and sights, nor will it allow you to go on any walks or hikes around the island without wasting a ton of time. There aren't bus stops every mile, and between the schedules and routes, you'll quickly find that a lot of places just won't be accessible to you.

Love seeing all the positive and exciting comments about Santorini. It’s well worth visiting multiple times.