As you might expect for an idyllic Mediterranean island paradise, there's not a huge amount to do on the island. In the summer this is just as well as it gets really hot.
In AvÅa there's a "Lunapark" fair. There's also sandy beaches all around the island, some public, some private. There are a few real gems with some natural bays due to the island's shape. It's possible to walk over the main hills fairly easily, but make sure you take plenty of water if you do and either set off very early in the morning, or in the mid-afternoon.
The main strip in town might look at little tacky, but if you go up to the ferry port, nearby there's a pair of fish market stalls. The guys there are more than happy to show you how they handle the fish, the fish are fresh and great cooked in a Turkish barbeque mangal.
Heading down the strip and out you'll come to a place called Mavikoy. Before heading up the hill there's a restaurant with a sea view, some nice beaches and past the hill is a lookout point sometimes frequented by couples around sunrise/sunset.