I’d say there are only 2 reasons you’d ever come to Jogja as a tourist. One is to see Prambanan and the other is to see Brobodur. Prambanan is a Hindu temple and is about a 15 minute drive from downtown depending on traffic. It’s easy to get to and stunningly gorgeous. You could almost make the argument that if this were any other city, Prambanan and Borobudur would each be their own major attraction.
Getting here is easy either via taxi or public bus. You can negotiate a flat rate with a taxi driver who will take you there, wait, and bring you back. The cheapest option would be to take the public bus but you would need to make sure you understand that bus times may not always be accurate (if ever). For 300K rupiahs (about $20-$25 USD) I just hired a taxi. This would obviously be entirely up to you but since Jogja is a fairly cheap city, I didn’t really feel the need to sacrifice speed and comfort for price. Considering it costs me $20 each way to get to the airport from my house in San Diego, I wasn’t going to quibble for someone to drive me.
I’d STRONGLY urge you go visit ALL tourist sites in Jogja on weekdays vs. weekends. Tourist sites are incredibly popular with the locals and regardless of what time you go, expect the weekend to be completely packed with people. If the weekend is unavoidable, go early. Prambanan is popular for people to go watch the sunset so if you want fewer crowds, go in the morning or earlier in the day. I went on a Sunday but got there early enough that there weren’t many people when I showed up but by the time I left, the place was getting pretty crowded.
The park itself is fairly small so I’d recommend taking your time to walk around. Also, the main temple complex is Candi Prambanan but I’d recommend you walk to the far back temples called Candi Sewu. It’s about a 15 minute walk and you can walk through the ruins and run into far fewer people.
I’d also recommend getting the combo pass for both Prambanan and Borobudur. I think it was around 550K rupiah and you can pay via credit card or cash. The catch is that you have to use the ticket on consecutive days and it does NOT provide entry to sunrise at Borobudur. However, you can save some money and unless you’re going to be staying here for an extended period of time, I don’t see any reason why you wouldn’t go on consecutive days.