Akureyri is a very picturesque town with its location by a scenic fjord, eclectic building styles and streets winding their way through gorges. Spend some time walking around the town centre, see the impressive church and check out Listagilið literally "the art canyon", home to a number of arts and crafts galleries and shops. The tranquil municipal botanical garden, Lystigarðurinn (http://www.lystigardur.ak...), is worth visiting, as are the several small museums dotted around town.
Note that Akureyri is perhaps picturesque by Icelandic standards, but one may question whether it is a tourist destination in its own right. Tourists to Iceland typically go there because of the splendid scenery, not because of the cities. Akureyri is primarily a local hub, a place to start excursions to more interesting places.