You can start by checking out which region of islands fits you more, for example the cycladic ones are drier with the typical white houses, while ioanian islands (Corfu, Lefkas, Kefalonia, Zakynthos, Ithaca, etc) are green with different style. Then there are the Sporades (Skiathos, Skyros, Skopelos, Alonnisos) and the Dodecanese (Rhodes, Kos, Ikaria, etc).
If you choose the Cycladic islands I would suggest Milos which is my favourite for great beaches, food, landscapes and relaxation, also Naxos has a lot to offer. If you don't go in high season (mid July- end August) then also some of the more crowded islands can be more relaxing, like Paros, Mykonos and Santorini.
If you choose Ionian, then Ithaca is the most relaxing of all, but you can also chill-out on Kefalonia and Zakynthos if you stay away from the 2-3 villages that are more touristic, eg stay away from Laganas in Zakynthos.
Make a search to decide towards which island group you are leaning, there are a lot of experts in the relevant forums to help you with more specific questions.