There are more than 6000 islands in Greece and it is very difficult to answer to this question. Classified according to the beaches, the sea, the historical places and activities, here are a few islands that you have to visit.
For the most beautiful beaches: Mykonos, Crete, Kefalonia, Skiathos and Lipsi
These islands have beautiful beaches which all have their own characteristics: the Venetian palace of the Frangokastello beach in Crete, the turquoise sea of the Platis Gialos beach in Lipsi, the creeks of the Myrtos beach in Kefalonia, the 65 beaches of Skiathos and the legendary Super Paradise beach in Mykonos.
For historical places: Rhodes, Corfu, Delos and Patmos
Rhodes has an old city that will delight history fans. In Delos, near Mykonos, there are ruins of temples dedicated to Apollo. Corfu Island has monuments in honor of the First World War soldiers. On the island of Patmos, you will be able to visit the ancient monasteries.
For gastronomy: Corfu, Crete, Lesvos and Samos
The island of Lesvos is famous for its olive oil and ouzo (liquor). You will find typical Greek specialties and dishes which are influenced by Turkish. All these islands offer seafood and fine wines.
For diving: Karpathos, Paros, Milos and Kastelorizo
The diving school of Karpathos is renowned for its diving organized in the underwater hidden coves. In Milos, you will be able to swim with dolphins.
For a walk: Naxos and Skopelos
The island of Naxos has old trails that are perfect for hiking and walking. Skopelos is an island covered by forest and flowers that you will be able to contemplate during a walk.