You can rent a yacht with your friends or family and take a daily cruise to Oinousses island. Explore the small island, swim, have lunch and get back to Lagada village in Chios. (info: http://www.chiosyachting.gr)
You can have a demonstration of all of the stages of mastic tree cultivation and try yourself with traditional tools to harvest the mastic. Every Tuesday and every Friday in July and August. Mesta village (info: http://www.masticulture.com)
Take an easy walk on footpaths where depending on the season you can identify orchids, herbs etc. Every Friday excluding July and August.
Take a guided tour with a professional guide. Learn about every place you visit with accuracy.
Try a different beach each day. Especially in south Chios you can find unspoilt beaches that you can explore and enjoy. (Vroulidia, Mavra Volia, Kato Fana, Salagonas, Apothika, Ag.Dynami, Elinda, Trahili, Managros)