Between the churches Agios Markos and Panagia Erithiani|9, Agiou Markou 1, Vrontados, Greece | Map
Look for Homer's Stone
It's a rock shaped somewhat like a throne where the eponymous poet supposedly sat, composed, and performed some of the original masterpieces of Western literature.
Drink some mastika
The local drink is a sort of clear brandy that is flavored with resin from the mastic tree and has a taste reminiscent of licorice, and a kick like a mule.
Sit on a lovely porch in the sun and enjoy sfougato, a meat pie that's very popular in this corner of the Mediterranean.
Get into Classical and Byzantine Greece with visits to Chios Archaeological Museum and Byzantine Museum.
The fastest way in is via a short hop flight from Athens or Thessaloniki, but you could also take regular ferries from Samos and Piraeus. In addition, different European carriers offer flights to Chios during the high summer season. If you're in Turkey, there's a daily 11am ferry (one hour) from Cesme to Chios.
There are over 100 hotels on Chios, so you're spoilt for choice. Be aware that booking in advance is highly recommended though.