There are two daily Ethiopian Airlines flights between Addis Ababa and Dire Dawa (US$143, one hour), one continuing to Jijiga (US$49, 25 minutes) and the other to Djibouti (from US$80, 50 minutes).
A few regular buses depart daily to Addis Ababa (Birr158, 10 hours) between 4.30am and 5.30am from the bus station. For better comfort, use the daily Sky Bus (Birr275) and Selam Bus (Birr305 to Birr325), both with 5am departures and ticket offices in front of the train station. Stopping anywhere along the way requires paying the full Addis fare.
Ee City operates a daily bus to Gelille (Birr185) at the border with Djibouti, where you'll find onward connection to Djibouti City. It leaves around 7pm.
Minibuses run every 10 minutes or so to Harar (Birr21, one hour).
Car & Motorcycle
Trains to Djibouti City resumed in 2017. The new railway station is about 11km north of the city. From the old railway station, there's also a twice-weekly service to the border town of Gelille (Birr155, seven hours), from where you can catch a minibus to Djibouti City.