Kos is a lovely island which has been taken over by tourism.
Most of the nice beaches have been organized or serve resorts and package tourism (whenever the weather permits) and not just during high season.
Early June is probably the best time to go to Kos to avoid large crowds.
The island has a lot to offer including a cultural mix of Turkish and Greek.
Its main attractions are the lovely main port of Kos town with its Knights Hospitallers Castle, its ancient ruins and its Turkish quarter.
Kos is also famous as the birth place of Hippocrates and the Asklepion of Kos an ancient healing sanctuary visited until late Hellenistic times. The island is also beautiful and nice for walking especially in the less trammeled central portions of the interior. Also see Eco-Kos Guide