Icaria is probably one of the most beautiful Greek islands. It is very mountainous and at some points quite inaccessible, with lovely, green valleys, wild falcons and some amazing beaches.
Although not a very touristy island, you can get basically everything here: water sports, nightlife, shops and car rentals. The people on the island live off fishing and farming, and many work on boats or are involved in tourism. The Icarian wine has been produced and praised since antiquity.
This is where Icaros fell and was buried, according to mythology, after flying too close to the sun when escaping from Crete with his father Daedalus. Since then the island was named after his name Icaria and the sea around it the Icarion Sea.
Icaria is an island worth exploring, since it has some really nice little villages and beaches.
In Agios Kirikos there is an archaeological museum.
Therma is a health resort with radioactive hot wells, that many visit to be cured from various diseases such as rheumatism or childlessness. It can get quite busy here, since many Greeks from all over travel to Icaria solely for the baths.
The monastery Evaggelistrias is active, and open to visitors. Just make sure you are dressed appropriately: covered shoulders and long skirts for women, long trousers for men.
In Kambos there is an archaeological museum with various artifacts from the excavations in the area. Here the ancient capital was called Oinoe ("Wine") since the wineproduction was important here.
There is also an old monastery called Theoctistis ("God-built").
In Nas you can see the ancient ruins from a temple dedicated to Artemis.