Harput — really part of Elazığ but easy to treat as a day in its own right. Castle, Ulu Cami, Arap Baba, the village lanes.
Hazar Lake — 45 minutes south for swimming, a fish lunch and a striking tectonic landscape.
Keban Dam — north of the city to one of Turkey's largest reservoirs and the island-castle of Pertek.
Malatya — 100 km west, the modern city is unremarkable but the rock-cut churches and Hittite sites at Aslantepe and Battalgazi are worth a day each.