Egilsstadir is the large town in the Eastern part of the country even if it does not have more than 4 000 inhabitants! I like Egilsstadir as my closest Icelandic friends live over there, but, aside from this, it is especially pleasant to go on a car itinerary down route 931 which runs alongside Lake Lagarfljót located not far from there.
From Egilsstadir, I like to go for a walkin the main Icelandic forest, walking up to the beautiful Hengifoss waterfall and stopping at the museum dedicated to the writer Gunnar Gunnarsson where I watch the progress of the archeological excavation of a 15th century monastery. It generally takes me at least 1/2 a day to do all this.
If you want to go farther, do not hesitate to visit the very charming small village of Seyðisfjörður which is enclosed within the fjord.