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 Listagili├░ literally "the art canyon", home to a number of arts and crafts galleries and shops. The tranquil municipal botanical garden, Lystigar├░urinn (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.