The currency of Indonesia is rupiah (IDR). You may cash your cards in many mini-banks/ATMs scattered around the city. Modern stores normally accept payment using major credit cards.
Public transport (bus) is not so good in Yogyakarta, but instead you can conveniently using car taxi or motobike taxi, if you like. You may download mobile applications for using their services. For car taxi, for instance, you may use Say Taxi (Android, IOS), and for motorbike taxi you may use GO-JEK (Android, IOS). You can even use GO-JEK for ordering meals and doing groceries.