Getting around the city is actually not very hard today. I guess 10 years back it was a complete different story, the Becak and the Andong plus the bycicles where probably populating the entire city, however today you will most likely see bikes, thousands of them and cars, more and more cars everyday.
Your options are many:
Rent a bike: One of the most chosen ways of transport in the city is to rent a bike (A motorbike) and just drive. I would dare to say that 90% of the expats in the city drive one without a license or a license from their country. While I will advise you to make one as fast as you can, if you plan to stay for a little while this is a good option.
The price range for a 110cc bike (Scooter) goes from 400k to 600k IDR a month. Daily you can find worse deals, like 40k-50k.
Rent a Car: The city is normally packed and whatever trip you want to do by car, always add it 20 minutes to what it would be by bike. Still car is safer if you are not confident to drive around the city, or if you have a family. Also probably if you have to transport a tribe of penguins as they wont fit in any bike.
The options in Jogja are many and the price range will depends a lot on the type of car, the features of it, like TV with DVD player, etc. It also comes in Jogja with your own private driver. Something you will never imagine to afford in your country but here in Jogja it’s all possible. Anyway yes, the prices are affordable in general.
Of course, it goes without saying, you can always buy your own car and it will eventually be cheaper. (The drivers are not for sale in 2018, perhaps 100 years ago they were).
Ride on demand: If you dont want to drive, or you can’t afford a car with a driver, there is always an option in Jogja, actually several. Today most of the people rides with different mobile applications.
Go-Jek: You will see the drivers all around the city, they are wearing green.
Download the app in your Iphone or Android and start riding. You can choose among bike drivers or car drivers, it’s clearly described in the app as GO-RIDE or GO-CAR.
The only issue is that they always call you to confirm the pick up point and that it’s mostly in Indonesian. Try to always have a translator next to you when asking for a ride or you will bump into a lot of frustration
Price range goes from 5K the kilometer.
Uber: Does it need presentation? Download it and start using it. If you have an account in your country it will still work but you need an indonesian sim card to be able to use it, but still the same account and even the same credit card.
Grab: If nothing convinced you, then you can try Grab, is like Go-Jek but with another name. Who knows, maybe you want to try something new? Also available in all platforms.
All apps above described have a very intuitive system, choose the pick up point from a map (You can choose to use your own location) and then just pin in the map again for the drop off. Watch out for streets non-existing. Jogja is currently booming and still many new streets are not in the maps. Be patient and ask the driver to drop you close from there, then you can just indicate him or walk.