10 days in Greece itinerary without a car

Why Hydra instead of Paros or Naxos?

I can always change that, to be fair, I have no clue which islands are best to relax, the only place I know are Mykonos and Santorini from other travelers. But feel free to make any modifications.

Substitute Paros or Naxos for a better total experience.

So the only modification i should do is:
Day 6: Morning bus to Athens, then boat to Mykonos
Day 7: Relax in Mykonos, Sleep in Mykonos
Day 8: Take ferry to Paros or Naxos, sleep in Paros/Naxos (depending on my choice)
Day 9: Relax in Paros/Naxos, Sleep in Paros/Naxos
Day 10: Take Ferry to Athens and sleep there.

We flew in to Mykonos from Milan and it was the only reason we went there. It was ok, more expensive and very commercial . We then ferried to Naxos and Paros for 4 nights each and loved both islands, ferries and the people. We also enjoyed our ferry to the port for Athens. We had alot more time and visited other places. Hydra was nice but not the same as either island and took alot of time to get there. Athens is very interesting and we totally filled our 3 nights there. Like everyone advises end on Athens to make sure you don't miss that plane home. There was a national 2 day strike when we were in Naxos and alot of lines and nervous tourists. We lucked out by getting on a tour boat from Naxos right to Naroussa, Paros. I don't see what month that you are travelling. We were there in early September and love, beachs and swimming especially in Paros,

Many experienced travellers choose to leave Athens for the end of the trip. You need to be back in Athens the night before your return flight so it makes sense to do the Athens portion just one time.

If it were me I would get out to your first island the day you arrive in Athens. Is there a particular reason that you have chosen Mykonos as your first island?
It has branded itself as a high end destination with prices to match.

You have planned 4 nights on 2 islands. Instead I would go to Naxos, Paros would also be a good choice. My preference though would be Naxos. Depending on your budget and timing you might consider booking a flight to you island choice when you arrive in Athens. You need 2 hours to make a connection. Start your Greek adventure on a relaxing stress free Island. Spend 5 nights.

While island hopping sounds like fun in fact it gets old very quickly. Every ferry hop essentially eats up half a day of your vacation and you are planning 3 hops which means you lose one and a half days on the ferries.

You could fly or ferry back to Athens and if the bus schedule fits get to Delphi on day of arrival back in Athens. Spend your 2 nights Delphi then return to Athens for 3 nights before your return flight.

The logistics I have suggested actually gives you an extra night on an island.

Here is why I choose Naxos
Around Naxos town 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299959098
Back to Naxos 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299940917

I only chose Mykonos because it's the only island along with Santorini I know and I'm willing to modify it. It's a good idea you have to start my journey elsewhere.

I also spent a night in Delphi.

My only experience on a Greek island was 2 nights/most of 3 days, in Crete. I would have been as happy if I had skipped Crete and seen Olympia in western Greece.

Consider leaving Athens right after your plane arrives; putting all your nights in Athens at the end of your trip. Consider taking the bus to Delphi the day your plane arrives and spend your first night in Delphi.