Eskifjörður is my favorite town in the east because they have the best ski resort in Iceland in my opinion. The ski resort is a little bit out of town and up the mountain, it’s called Oddsskarð if you want to check it out. Another reason to go to Eskifjörður is that they have the best swimming pool in the east fjord.
In Eskifjörður you also find a museum about fishing and some beautiful old house along the harbor. Really nice little town.