week three,,,,,Islands (but which)

Before you eliminate Mykonos, google Delos which is just a short boat ride from Mykonos itself. It has been called "The Birthplace Of The Gods" and "The Home Of The Gods". Absolutely outstanding look back at history.
You may also be able to get a day tour to Delos from Naxos.

RC, in 2017 I took the excellent RS Best of Greece guided tour, then continued on similar to what you suggest, flew to Santorini for 3 nights, then short ferry ride to Naxos for 3 nights, then flew back early morning to Athens, for a final day and night before flying home. Worked great for me. Stayed at the modest Villa Fotini in Firostefani on Santorini, good location and affordable, took an excellent tour of the island by bus one day, spent another day exploring by public bus. Really enjoyed Naxos, not tourist-dominated like Santorini. Stayed In the top-floor apartment at Santa Katerina Apartments, nice comfy spot in a good location. I arranged a private guide with car to explore the island one day. Both these islands have much to explore beyond the most heavily touristed areas.

Daytrip excursion boats to Delos are available from both Naxos and Paros.

You seem to have a pretty good plan. Many experienced travelers to Greece recommend flying to their first island. Generally they head to the furthest island then ferry hop back towards Athens. Many prefer to fly back to Athens (or Ferry) for the last few days of the trip so they are in Athens for the return flight home.

I honestly think that every first time visitor to Greece should try to get to Santorini. It is expensive and crowded but those views are amazing. You will still be in early season so cruise ship crowds probably won't be much of a factor.
Its a 2 hour ferry trip to Naxos. We always take the first fast ferry (generally one of the Seajets large fast boats. The first one of the day will have you in Naxos by noon. One of the bonuses is the Championjets have an outside deck so you can see the Caldera departure from the deck.

Naxos is a great companion island to Santorini. Its close by, it has everything you might want, blue trimmed white buildings, a lovely main town topped by a Venetian era Kastro, mountina villages a smattering of antiquities, great beaches and most of all a local population who are pleased you have come to their island.

I know you said you are not looking for a beach holiday however I still would recommend you find a hotel at St George beach. It is part of Naxos town but is only a 10 minute stroll into the town. In the early season the tourists will be mostly staying in this area. You have easy access to the transportation hub to explore the island.

Here is what you will be seeing in Santorini and Naxos.
Santorini http://www.flickr.com/photos/stanbr54/sets/72157632154800782/
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/

Trip around Naxos http://www.flickr.com/photos/stanbr54/sets/72157634605629689/

Naxos trip 2019 https://www.flickr.com/photos/stanbr54/albums/72157711294807761

If you would like to read about our experience in Naxos in September here is a link to my trip report.
https://community.ricksteves.com/travel-forum/greece/trip-report-nine-days-in-naxos

In particular take note of the section of the report where we took a private tour of Naxos specifically a visit to a goat farm and cheese making operation way out in the country. This was a small tour with max of 7 people. It included a meal at the farm kitchen, we helped make cheese, fed the goats and then went to a winery for a tour.

In case you don't want to read the whole trip report here is the excerpt on the farm/winery tour.
This is how we found Philema tours. It turns out this is a new company. The owner Elaini, was a tour guide for Naxos tours but when she discovered the bus driver was getting 200 euros to drive the tour bus and she was getting 30 euros per tour she quit and designed several unique itineraries that included local farmers, wineries and olive oil producers. This is a small tour with a maximum of 7 people plus Maria and the driver. For this trip we had one other couple with us. They were from New Orleans and were convinced they didn’t have any accent at all.

The farm was way out in the country in an area we have never visited before. As soon as we arrived they herded about 20 goats into a nearby pen and we were allowed us to feed the noisy, rambunctious crowd of goats and play with their farm dog Harry. Harry was quite put out that the goats were getting more attention than he was. This close encounter with goats allowed us to ask why some goats have a bell while others didn’t. We thought that the most intelligent goat was a natural leader and the other goats would follow the bell. It turns out, the naughty goats get the bells so the farmer will know where they are at all times. Naughty goats climb trees, are escape artists. Climb fences, butt people, dogs and anything else that’s handy, and are always pushing to the front of the food trough. Lolly has since been threatening to hang a bell on me when I am not on my best behaviour.

We moved inside and were greeted with a big farmhouse table laden with food. There were spinach and cheese pies, zucchini fritters plus two egg and cheese dishes bread olive oil and homemade wine.
We discovered the only way you can get a Greek host from refilling your wine glass was to leave it full when you have had enough. I think I learned that lesson a bit late in the process but I wasn’t driving.
Behind the table was a large caldron of milk on a heater. This was fresh milk from the goats which had been simmering on the heat for the past two hours. The farmer brought out a stick and started to slowly stir the milk and it began to lump up. They had plastic baskets/molds and the farmer began to add the curds to the baskets. Each of us got an opportunity to fill the baskets and pat down the curds into a solid mass. Each basket was then submerged in the Whey and the cheese was coaxed out of the molds turned upside down and reinserted into the mold. The cheese is stored in a separate room in coolers and is aged for several months. Apparently the little local products store where we buy spices, wine, ouzo Kitron and cheese is actually the business that purchases the cheese from this farm. That made our day, to discover the cheese we made today, would eventually end up in a shop we know well.

The tour next went to a winery. Apparently the wind in Naxos makes it very hard to grow grapes but what survives makes a high quality wine. We toured the grape field then were taken to a lovely sheltered picnic area where we had a wine tasting of 5 different wines. I am estimating that each pour was at least 2 ounces. Then they brought out two bottles of their home made wines.

Spent 3 nights on Santorini (Oia) in April 2016 and really enjoyed it. Views were great and the crowds light.

Phase One - The RS DIY car routes are usually fast-paced. They are often the same pace as the RS guided tour, where a veteran bus driver handles the road, and mostly gives you door-to-door service (hotels, sights, etc.). You will have to navigate the unfamiliar roads (possibly with limited signage in places) and find all the sights, hotels, and parking on your own. Nor will you have a guide to take you from place to place and usually skip the lines at sight entrances. You will want a day or two in Athens to get over jetlag before renting the car, and you'll want at least one night there before your flight home. If you are interested in archaeology, there's definitely enough for 3 full days in the city.

In October I spent a week on Crete and a second week split between Santorini (3N) and Naxos (4N). I was on Santorini right around mid-month and although it was end of season, there were 1 or 2 large cruise ships tendered off Santorini every day. I arrived by ferry from Crete and it was nearly an hour to get to my hotel in Firostefani (on the outskirts of Fira) by private car. The driver told me it was his last week of the year, there'd by no work after that. Still there were 1 or 2 large cruise ships in port every day I was there. I found a budget place to stay and chose Firostefani because it was walking distance to Fira but without the crowds. I thought both Fira and Oia were very touristy. The classic views in all the photos are from Oia. Instead of going early in the morning, before the cruisers, I ended up arriving around 10.30. While it wasn't overly crowded, there were lots of people and way too many tourist shops. So I was able to enjoy a bit of the "classic" views and a couple of okay sunsets in Firostefani - reputed to be just a tad less than those in Fira - and they were not nearly as wonderful as I'd expected. Santorini was the most expensive place for food and drinks by far - even more so than Athens. In all I came away thinking it was mostly hype. If you are willing to rent a vehicle and avoid these highly touristed areas, I think you can have a very enjoyable time for 3 nights. Because Santorini is so popular, both the ferries and the airport are often very crowded and disorganized. Allow enough time to arrive and leave.

On the other hand, Naxos was a great place to relax. I stayed in a small apartment on St. George beach. It's very shallow water so it attracts many families but not in October (or, presumably in April). It was walking distance to the center of town and to a couple of very good restaurants. I took a full day guided island tour that was interesting.

By far I enjoyed Crete the most. I spent 5N in Chania and 2N in Heraklion. Chania is charming, not too touristy, has interesting sights and some great places for late afternoon cocktails and great dinners. I stayed in Heraklion in order to visit the ancient site of Knossos and the archaeology museum. It is possible to do both in a single day, but I wanted a very slow pace for this trip. If you rent a car, you could easily spend 2 weeks just on Crete. There are direct flights between Chania and Athens. To get anywhere else, by plane or ferry, you need to go from Heraklion.

We also traveled the Peloponnese and then spent time on the islands, so I would recommend Santorini and Naxos. We were there longer than you, so Crete was also a part of our trip, but you do not have enough time to include that island. Yes, Santorini is expensive and filled with tourists, however IMO it is magical. We stayed in Firostefani at Lithies Traditional Homes and we chose the "Eros" home. I wanted a place with a view of the caldera and Lithies was the perfect spot. The service from Maria is exceptional; Eros was well appointed and sunsets were awesome. We felt it was a great value for the price. One evening was spent in Oia and it was insanely crowded with tourists, so for us, Firostefani was a better choice. We did take a catamaran tour one day, but I'm not sure if the weather in April would be conducive for that. Naxos was more laid back and since we were there in October, it was not at all crowded. I believe it is the largest of the Cyclades, so I booked a private tour with Nicolas at Naxos Discovery Tours. https://www.naxosdiscovery.net/ It was reasonably priced and we saw things that you will not see on a bus tour. Highly recommend! On Naxos we stayed at Hotel Grotta. It's small, family owned, a 10 minute walk to town and has the best staff and breakfast you can imagine. Our room had a small balcony with a view of the sea, so evenings were spent enjoying the sunset with a glass of wine. The hotel also has a very nice roof garden for those guests who do not book a sea view room. They also provided us with complimentary transportation upon both arrival and departure. We did take the high speed Sea Jets ferry from Santorini to Naxos. From Naxos we flew back to Athens. Enjoy your trip!