Tinos

Tinos is home to the most important church in Greece, The Panagia, and its miraculous icon. Pilgrims come from all over, day trippers just for a visit. There are some spectacular villages, many windy uphill roads, - I don;t know about the bus routes, I took my car, and generally found the roads a little difficult. It is also home to one of Greece's most renowned sculptor, Halepas.
It has many sandy beaches, most of the nice ones are accessible via water only, or dirt road. Its cuisine is not worth noting, and it is one of the hardest hit islands during the meltemi season [windy season]
It doesn't have much of a nightlife [mainly due to the religious attraction], but it is not as touristy or more authentic, if you like, than many of its neighbours.
This site has a good report on all its attractionshere