Excluding those in the Saronic Gulf, since I believe Hydra is part of the Rick Steves tour, the most easily accessible and best-known islands from Athens are the Cyclades. Those 22 islands are listed here:
https://www.visitgreece.gr/islands/cyclades/
9 or 10 days is a good amount of time to visit two islands, provided they are connected by ferry routes. So if you want to visit two islands in the same trip, first check how you can travel between them.
There are several ways to determine which island you want to visit.
If you're bothered by searching, just look at the most touristy and easy-to-combine islands, which will certainly be recommended here. These are Santorini and Naxos (which have airports). There's also Naxos and Paros, which is another easy pair to combine.
I don't recommend going to Mykonos unless you want to party, see celebrities, and spend a lot of money.
The other approach is to know why you're going to this island rather than another. I've visited many Greek islands, but I've never chosen an island without knowing why I chose it over another, and without considering what I could see/do there. So there are quite a few other possibilities if you do a little research.
Also consider transportation. If you really want to explore an island, a rental car is essential, and you'll need at least four or five full days. If you don't plan on renting a car, be aware that while some islands have a good bus network, others have almost none.