Jogja or Yogyakarta n els...

Yogyakarta is somewhat a historical city. It has been the capital city of the Mataram kingdom, and around 1949, it was the capital city of the newly formed Republic of Indonesia. So you will definitely see palaces here. I noticed three palaces in Yogyakarta, no, make it four. First, the Presidential Palace (Gedung Agung), which is now still used for Presidential ceremonies or activities in Yogyakarta. Second, the Kraton (Sultan's Palace). Historically, Yogyakarta was a sultanate, namely Kasultanan Ngayogyakarta Hadiningrat. The governor of Yogyakarta until now has always been Sultans of Yogyakarta. These two palaces are not far from each other, both are near Malioboro. And these two palaces are still used as 'real' palaces, not for exhibition only. Around Kraton, you will also find a place called Water-Palace. It was a beautiful swimming pool for the royal family. It is not so beautiful now, but you can still imagine it, though.
The next one is the Kraton in Kotagede (Kraton actually means king's palace). Kotagede is in the southern region of Yogyakarta city. The kraton here is older than the kraton downtown, and not as big either. Kotagede was once the center of the kingdom, but not now. In Kotagede, you would probably rather prefer to see silver handicrafts instead, because Kotagede is a silver handicraft village. You can see the workers carefully making beautiful silver handicrafts at the workshops. The fourth palace is Kraton Ratu Boko (Queen Boko's Temple), which i think is the oldest palace of them all. This one i visited for the first time just a few months ago, it isn't as famous as other places in Yogyakarta. And, this palace is not actually inside the city, it is outside. (but not so far though)
Kraton Ratu Boko is a palace built on top of a hill. It is near Prambanan temple, so i recommend you visit both at once. Kraton Ratu Boko is still 'new', the government has just renovated and recovered the site just a few years ago. It has typical Kraton structures, it has a swimming pool, it has a 'hall' (an open one though), it has a gate, just about everything. It is very exciting to see this palace, because you need a high imagination to 'reconstruct' what you see there to what it probably looked like originally. I mean it is severely ruined now, but still interesting.
There are two kinds of temples in Indonesia, Hindu temples and Budha temples. I think Borobudur is a Budha temple (one of the largest and most famous in the world). Although i am not a Budhist, I highly recommend this place. Leaders from other countries even come to this place. So, it's a must.
Borobudur is not in Yogyakarta, but to get there from Yogyakarta is not a problem. The journey nowadays might only take about one hour.
Prambanan temple is closer to Yogyakarta. It is the second largest temple I think, and unlike Borobudur, it consists of several hundres of small temples, except few of them are large and tall temples. A tale mentioned that there should be 1000 temples to be built at Prambanan, but only 999 was built. This is only a tale.
Surrounding Prambanan temple are villages and rice fields. Among these rice fields, there are actually many smaller temples. These are Candi Plaosan (candi means temple), Candi Sewu, Candi Kalasan, and many-many more, i can't remember them. It looks so natural and traditional, you would probably forget that you are now in the 21st century.
the Ramayana Ballet which is shown at the Prambanan Temple complex at night, especially on full moon a similar show downtown at Purawisata. It is a traditional drama performed only by dancing and no speaking.
There is one more temple complex called Dieng, and it is quite far from Yogyakarta. Located at a very high altitude, it is very beautiful with its colorful natural lake nearby. You'll probably spend one day for visiting this place. (mainly because of the long journey from Yogyakarta)
From the city, you can see a large mountain on the north, it is actually an active volcano, Merapi. To get a closer look at Merapi volcano, you can go to Kaliurang. It is a small town, which the former Dutch colonialist had made as a place for relaxing. In Kaliurang, there is an observatory post for observing the activity of the volcano. Kaliurang is also surrounded by a natural rainforest protected by the government. It is cool up there.
At the opposite direction, at the south, you will meet beaches. The closest beach is Parangtritis, but it is not so good. The waves there are enourmous and highly dangerous. The beach itself is so long, there are beautiful hills nearby, but it just isn't so good for me. Just see it for yourself and you will understand what i mean. I prefer more natural beaches in the Gunungkidul region, like Krakal, Baron, Wedi Ombo, etc. They are far (some are very far), but it is worth to see.
The most famous food in Yogyakarta is Gudeg. I can't explain it much to you about it, basically it is brown, and it is sweet, it has some eggs on it too. Then, about souvenirs. For those who are interested in batiks, buy some at Beringharjo Market in Malioboro. It is much cheaper there. But for more elegant and expensive batiks, find them at batik shops along Malioboro street.