Travel to greece

If you only have 2 weeks, by deducting the arrival and departure days, you will only have 12 days left.
Keeping 2 days for Athens,it's only 10 days left to go to the islands.

Since you could spend those 10 days in Crete without seeing half of it, try to limit yourself to Crete and Naxos.

Do not wait to look for accommodation if you will be 10 travelers. Mid July to mid September is the busiest time of the year.

Follow the usual advice to find a connecting flight to Naxos or Crete as soon as you arrive in Athens and save Athens for the end of your stay.

Forget Hydra which is only of interest for people who don't have time to go to an island and who are doing a daytrip from Athens

Try to fly in and out of Crete and Athens called multi-city. I would focus on Crete and Athens and forget Naxos which is best if visiting the Cyclades Islands. You need two weeks for Crete if not a month.
The other alternative is to forget Crete and focus on the Cyclades and Athens. This means you should fly in and out of Santorini/Mykonos and Athens. Santorini and Mykonos gets hordes of cruise ship passengers so you will be walking shoulder to shoulder with people during the day. Naxos sits in between Santorini and Mykonos.
I agree with JoLui forget about Hydra that will also be unbearable hot because there is no wind during the summer.

We had ~2 weeks in Greece from 9/19/22 to 10/3/22 {shoulder season]. We spent 5 days in Athens, 6 days on Crete and 3 days on Santorini. Would go back to Crete and Athens in a heartbeat! Even with 6 days on Crete, we still have so much to do! We stuck mainly on the east side and look forward to exploring the other side. Santorini was fine, but really have no reason to go back - very touristy and overpriced even in October when we were there. Loved traveling by ferry from Crete to Santorini. In Athens, we did a foodie tour and the tour to the Acropolis with the museum add-on. The museum was very much worth it, as was the guided tour to the acropolis. Sign up for the earliest morning tour you can as it will be HOT otherwise. Good luck!

I think you have two options, fly to Crete immediately on arrival in Athens. I would fly to Chania there is one flight that gets in around 9pm. If you can't make that connection then you can fly to Heraklion. Overnight there perhaps visit the archaeological museum and perhaps Knossos and then bus to Chania.
Crete is very large with 14th century, cities with old towns and harbours. Because there is so much to do and see west Crete makes a lot of sense. Just concentrate on Crete.

Or if Santorini is a bucket list must do, fly there when you arrive in Athens. There are late night flights to Santorini so getting a connecting flight should be doable from Athens.
While it is expensive and crowded particularly when cruise ships are in town you can manage the crowds by looking for a hotel in Firestophani which is a suburb of Fira the capital but just far enough away that the crowds do not go there. It also has hotels with Caldera views.

There is a reason the cruise ships come and why its crowded, because the views are spectacular.
3 nights is all we can afford before we make an island hop.

Santorini 3 nights plus a week on Naxos is a great combination. Naxos has been called Mini Crete because of its fine beaches, mountain villages, hiking, some antiquities, a main town with a waterfront promenade plus a maze of back alleys all leading up to the 14th century Venetian Kastro at the top.

Santorini and Paros also make a decent pairing for a ten day island hopping trip. Naxos though is our favourite.

Then fly or ferry back to Athens and do your sightseeing and touring of the antiquities right in the centre of the city.

Perhaps I can tempt you with some images.
Chania 2022. https://www.flickr.com/photos/stanbr54/albums/72177720299646015

Santorini https://www.flickr.com/photos/stanbr54/albums/72157632154800782

Around Naxos town 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299959098
Back to Naxos 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299940917

With at least 2 other islands already on your agenda, Hydra is unnecessary and a poor use of your limited time.

Do put Athens time at the end to facilitate your homeward departure.

Persons 6-10 need to decide soon so you can get to work on lodging. ( or accept everyone may not be in the same place)

It will be hot--- happy travels

August (esp Aug 15) is when all Italians are traveling, so the later you go the easier and more economical it will be.

August (esp Aug 15) is when all Italians are traveling

Not just all Italians, almost all Europeans, Greeks included

A trip to Greece for your birthday sounds like an amazing plan! Here's an itinerary suggestion that would allow you to see a variety of islands and experiences within your 2-week time frame, even though it:

Day 1-4: Crete

• Fly into Chania and spend a few days exploring this beautiful city
• Visit the Old Venetian Harbor, the Archaeological Museum of Chania, and spend a day at the stunning Balos Beach
• Take a hike through the Samaria Gorge National Park
• Visit the Palace of Knossos on your way to the ferry port in Heraklion

Day 5-9: Naxos

• Take a ferry to Naxos and spend the next few days exploring the island
• Visit the Temple of Apollo and the Portara (the entrance to an unfinished temple dedicated to Apollo)
• Explore the picturesque village of Halki and visit the Venetian Castle
• Take a day trip to the neighboring island of Paros, which has some beautiful beaches and a charming old town

Day 10-14: Athens

• Visit the iconic Acropolis and the Parthenon
• Explore the Ancient Agora and the Roman Agora
• Take a day trip to the nearby island of Hydra, known for its beautiful beaches and charming town

This itinerary would allow you to experience the variety of island types that Greece has to offer, from the largest island of Crete with its diverse landscapes and rich history, to the more traditional and laid-back island of Naxos. And you'll wrap up your trip with a visit to Athens and the nearby island of Hydra. Of course, this is just a suggestion!