Getting a cell phone in Ankara is easy because there are many many cellphone offices around the city. The ones you will see most often are Turkcell and Voda phone. Turkey has had problems with foreign cell phones and illegal use, and therefore you must have both your passport and your resident visa with you in order to register. Often it takes time to get your resident visa. If you need a phone sooner than that, find a Turkish friend to register for you. It will make no difference to the end result---doing this does not affect payment or customer services.
You will need to choose a phone and buy a SIM card to get started. Cellphones here are all prepaid and the minutes are called "Kontors". You can buy kontor cards for your phones at convenience stores, but they are more expensive. It is better to refill your minutes at an actual cell phone outlet. The cheapest cell phone on the market right now is a Nokia basic phone for about 70TL. If you are a student, go to Turkcell and get a student plan (TurkCampus Plan) so that your minutes won't disappear as quickly.