Akureyri punches well above its weight for a town of 20,000. The church at the top of the hill is the most photographed building in northern Iceland; the botanical garden, at 65°N, is one of the world's most northerly and genuinely impressive; and the art museum is small but well curated. The town is compact enough to cover on foot in an afternoon, leaving the rest of your time for the surroundings.
The real draw, for most visitors, is what lies east of Akureyri. Lake Mývatn, an hour's drive, is one of the great natural spectacles in Europe: geothermal activity, extraordinary birdlife, lava formations, and the blue Mývatn Nature Baths. Dettifoss — Europe's most powerful waterfall — is another 60 km beyond Mývatn. Budget at least two nights in Akureyri to do this region justice.
In summer, the Arctic Open golf tournament is played through the night in late June, under midnight sun. In winter, the ski area at Hlíðarfjall just outside town draws Icelandic skiers; it is one of the country's best facilities.