You do not have to worry about transportation that you can take when you are in Yogyakarta. There are so many kinds of transportation: becak (pedicab), andong (four-wheeled carriage drawn by two horses), GO-JEK or GRAB (ojek online), taxis, and Trans Jogja buses. For your information, the roads in Yogyakarta are busy, especially in business hours. Many Yogyakarta people ride motorcycles to get to their destinations. Some others certainly prefer to drive their cars.
Well-known offline transportation in Yogyakarta are becak, andong, and Trans Jogja. Becak and andong are considered to be the traditional type of transportation. Meanwhile, Trans Jogja is the kind of transportation provided by the local government. You can find either of these offline transportation easily in various spots of Yogyakarta (although, mostly, andong is found near tourism destinations).
Another option besides offline transportation is, of course, online transportation. There are two online transportation service providers: Gojek and Grab. You can book through their official applications in App Store or Google Play.