If you drive (or fly) to the North part of Iceland there are both activities and sights we can recommend you to do.
If we start with the "capital of the North" Akureyri, which is the largest town outside the capital area of Reykjavik. There are nice museums in town, such as the Aviation Museum, the Motorcycle Museum and Akureyri´s Art Museum. There are quite a lot of hiking and activities options in the area and for more information you can Visit Akureyri´s website.
There are good options in day tours from Akureyri, to Lake Mývatn area where you can visit (to name a few) Námaskarð, Námafjall, Skútustaðir pseudo crater, Dimmuborgir, walk to the top of Höfði peninsula, visit Grjótagjá and many other sights. Mývatn area is one of the most beautiful in Iceland and a day is barely enough 🙂
Also make sure you visit Sigurgeirs´bird museum which is the largest privately own bird collection in Iceland.
The Mývatn´s nature baths is the little sibling of the Blue Lagoon and well worth visiting for a relaxing time.
In a 40-minute drive north of Mývatn there is a town called Húsavík, often called the Whale Watching Capital of Iceland, which is a perfect stop to do a Whale Watching Rib Safari and visting their Whale Museum, before driving to Ásbyrgi. Ásbyrgi is a large horseshoe shaped canyon and one of the most beautiful and peaceful places in Iceland.
Hljóðaklettar is a place to visit in the summer time but not (easily) accessable during winter (see below)
From the end of September it is not possible to drive from Ásbyrgi to Dettifoss in a 2WD but during summer the gravel road is open for traffic of 2wd. To reach Dettifoss during winter you need to drive all the way back to Mývatn and drive a road that is open all year (but not serviced so in heavy snow it closes).