I haven't taken that train but I've travelled by train from Istanbul-Ankara, Ankara-Adana and Ankara-Izmir. Don't expect the train to arrive punctually but otherwise I strongly recommend it. I'm single and female and have never had any problems.

In addition to the Doğu Ekspresi you could also travel by the Güney Ekspresi or Van Gölü Ekspresi whıch both leave at 17:50 arriving at 11:50 the next morning. The Doğu Ekspresi arrives 01:30 so unless you're being met by friends this could be difficult. If you choose to travel overnight there are comfortable, affordable sleepers available with air-conditioning, fridge, basin etc and an adjacent dining car.

See here for further information: http://www.seat61.com/Turkey2.htm#Eastern%20Turkey
Or here for the Turkish railways network site in English: http://www.tcdd.gov.tr/tcdding/index.htm