One of the best activities is simply wondering around Parikia. It is a maze of shops, restaurants, and traditional homes. Straight from the dock leads into the maze. The Venetian-influenced buildings are so beautiful with their stark white outiside with vivid blue shutters and doors. In June, most of the houses were overflowing with flowers adding to the beauty of the place.
There are great shops in this area, offering traditional clothes, jewelry, books, linens and anything Greek you could imagine. The prices are actually comparable to Athens and I found some things were actually cheaper.
Unfortately, the greatest parts have to be discovered on your own and it is impossible to give directions to sights because it is such a maze. There is a bakery at the eastern part of the maze and it offers some good, if a little hard, Greek bread. Right across from the bakery is a good shop with Greek honey and herbs. The best thing we found is a church that overlooks the town. It looks directly east over the sea and every night we would watch the sunset. It was a perfect end to everyday. And you could quickly leave the overlook and find a cafe to start the perfect night with some ouzo.