2 weeks in June/July 2023 Greece

You can fly from Athens to any island but you'll need a ferry to get from one island to another, so you'll want two that are close to each other. The ones I've been to are Crete, Santorini and Naxos. I wouldn't recommend Santorini because of the crowds, though it may be possible to spend time on the much less frequented eastern side of the island - but you'll miss the famously picturesque parts. I loved 5 days in Chania on Crete. It's packed with history and un-kitschy shops. Heraklion has interesting sights and of course the ancient ruins of Knossos and excellent archaeology museum. Most ferries leave from Heraklion. The ferry connecting Crete and Naxos stops in Santorini. It's about 1/2 day to get to Naxos with its beautiful beaches and little mountain villages. Amazing food is everywhere.

I'm sure others will chime in with more ideas and recommendations.

With teens, I would suggest you consider Naxos and Crete. They are larger islands and seem to fit the bill for things that interest you. We had a rental car in Crete and were able to explore the island without having to rely on public transportation. We did not have a car on Naxos, but instead chose to hire a private guide for a one day tour that was phenomenal and checked your boxes. We loved Santorini for the iconic views of the caldera and your teens might enjoy a catamaran tour with snorkeling. If you do visit Santorini my suggestion is to stay in Firostefani rather than fight the crowds in Oia.

I would recommend Santorini and Naxos. I agree with the poster who suggests staying in Firostefani instead of Oia if you visit Santorini. And do rent a car for a day to see the rest of the island which is beautiful and very interesting, such as Akrotiri archaeological site, Akrotiri lighthouse, red sand beach, and black sand beach at Perissa. Santorini is crowded but the caldera views are worth seeing at least once in your lifetime.

Naxos is beautiful and more laidback than Santorini. You can relax on its beaches, visit the Old Town, and rent a car for a day for a roadtrip around Naxos. Visit the Temple of Demeter, the villages of Chalki and Apeiranthos, and the fishing villages of Moutsouna and Apollonas.

it will be hot so bring plenty of sunscreen and hats! And water when you are out sightseeing.

You can fly to Santorini, and then take the Blue Star Ferry from Santorini to Naxos. A 2-hour trip. Very easy.

You have received good suggestions so far.
What you have not told us is how long you will be in Greece. If its a week then you only have time for one island and Athens. If its two weeks then you have time for 2 islands and Athens. Three weeks would be better.

When you arrive in Athens immediately find a connecting flight to your first island. You need 2 hours to collect bags and clear passport control go up one flight from arrivals to departures and get through security.
If you choose Crete you could easily spend 2 weeks there and still only do one section of the island. There is so much to see and do there.

I believe that Santorini is a must do island. Its not authentic Greece but it is spectacular. Three days there will do nicely. Thats as long as my wallet can stand.

The combination of Naxos and Santorini is a winner. Naxos will give you an authentic Greek experience. It has it all, a great main town with waterfront promenade, narrow back alleys all leading up to the Venetian Era Kastro. It has good beaches, lots of hotels great dining experiences, Mountain villages, hiking and most importantly the locals are happy you have chosen their island.

Its essentially a 4 hour ferry trip back to Athens from Naxos. We choose to fly back in the morning so we can start our Athens experience by noon rather than late afternoon if you do the connection by ferry.

Here are some images of our trip. 3 nights Santorini, a week in Naxos and 4 days in Athens.
https://www.flickr.com/photos/stanbr54/albums/72157689937827955

And Crete.
Return to Chania 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299646015

Chania Sougia and Plakias 2019
https://www.flickr.com/photos/stanbr54/albums/72157711266062752

Western Crete https://www.flickr.com/photos/stanbr54/albums/72157632115757899

stanbr, relax -- the headline says they have 2 weeks for Greece (of course I always recommend a "three-weekend trip" if possible - leaving on a Friday to fly nonstop overnight, then coming back on the 3rd Sunday, thus getting 14 1/2 days On the Ground).

Oops. Thanks Janet, I must be getting old.

Because I just can't help myself--do consider the mainland. Nowhere else will you get the concentration of historical sites while sacrificing nothing in terms of natural beauty/sea/coast.

Be advised that Crete is huge and you really need a minimum of a week with a rental car just to barely scratch the surface. Take this into account when choosing your island destinations.

Your trip is more than 6 months away, why not try to do some homework by researching information on the dozens of other possible destinations in the Greek islands that you will find on websites such as:

https://www.aegeanislands.gr/

https://www.greekislandguru.com/

https://www.greece-is.com/

Staying in the Cyclades there is not only Naxos, Santorini and Crete. And even if you will not always be able to easily combine two islands in the same trip, see also Serifos, Tinos, Milos, Kimolos, Amorgos, Paros, Antiparos, Folegandros, Sifnos, Astypalea, etc.

Apart from the Cyclades you have about 200 other islands that can be visited

There are, of course, many islands that would work well but I thought I would share the itinerary we did with our young adult children in 2018 which really was nearly perfect. We spent 3 nights in Santorini, 4 nights in Naxos, 4 nights in Nafplio on mainland, and 3 nights in Athens. It was all of ours first trip to Greece. This last summer we returned and spent the bulk of our time (11 nights) on Crete.