Since you are coming in the beginning of January, there's a really good chance that it will be cold and rainy in the north. You'd have to go south (taichung, Tainan, Kaohsiung, Kenting) to find better weather.
Scenic beauty outside of the major cities: The rift valley between Hualien and Taitung. Taroko Gorge. Anywhere on the East Coast for that matter. The Northeast Coast. The Central Mountains. Kenting National Park... You can mostly drive or ride through/to these places, and/or walk/hike. Coming from Norway at this time of year you might really love the subtropical national park and beaches at Kenting on the very southern tip of the island.
Hotsprings: there are many hotspring locations around the island. Highly recommended. You can even stay in a hot spring resort hotel. If you go to Kenting, you could then scoot over to Chihben on the eastern coast for hotsprings. Take airplanes to and from Taipei...
Towns around Taipei:
Chiufen (Jiufen 九份), an old mining town in which much of the architecture has been preserved and turned into a lovely village of tea houses and pottery shops (was also the location for many films).
Danshui: Beautiful scenic area between two mountains at the mouth of the river that has now been overdeveloped. The renovated "old street" and "fisherman's wharf" are attractions.
Yingko (鶯歌) an old center for pottery that has been transformed into village with museum, old street, and pottery shops.
In Taipei:
the area around Dadaocheng (the old city), in particular around Dihua Street (迪化街) combines markets with colonial architecture.
Tea houses on th outskirts of Taipei known as Maokong (貓空) and also on Yangmingshan (a 1000 meter mountain at the edge of the city). These are great places to get a meal of fresh mountain vegetables. In the Peitou and Wulai areas, these are combined with hotsprings.
Since it's winter, try a restaurant in the city that serves Mongolian Hot Pot.
Hope this helps. I'm running out of steam.