Jogjakarta or as locals call it Jogja (pronounced Joja) is said to be the most culture rich city in Indonesia. I don't know about culture, but it's the perfect place to buy Indonesian arts and crafts - which are amazing and are sold for 5 times cheaper price than same things in Bali. Jogja is small (about 500,000 people), the city itself is nothing notorious except the sultan palace, but around Jogja there are many attractions - temples, volcanoes, lakes, wild beaches.
Jogja's central street, Malioboro is a bustling 1-2km long street market where you can find anything you want - batik art and art galleries, Indonesian wooden, leather and textile crafts, shoes, bags, souvenirs, traditional medicine, endless stalls with street food, pastries and sweet ginger "soups", various curries and veggies, roasted bats and wild pigeons.
We were lucky to try all that food before the celebrations of the end of Ramadan started. One day there were just no food sold in the city except crappy Dunkin Donuts. During this celebration most people don't work and have long holidays so consequently all local food places are closed. Even in our Sosrowijayan city area famous for its tiny streets packed with budget accommodation and restaurants - all places got closed during celebration.
The celebration itself consists of mass prayer next to the sultanate, and a parade the following day. During parade all mosque communities dress up, carry flags, beat the drums and carry mini versions of their mosques made from cardboard and decorated with lamps and lanterns. Some of them are human size, and some of them are barely 30 cm tall. Oh, and fireworks! Those guys sure love fireworks. For 3 days in a row fireworks started at dusk and continued till midnight. It was an amazing concert every evening - people cheering on the streets, occasional prayer on loudspeaker from a neighborhood mosque and endless fireworks.
In Jogja I also went for 2 full days to a Bahasa Indonesia (Indonesian language) school. I started studying it in Bali, and got my beginners level done in Jogja. The school is great, you study with private tutors who are all young and fun. Here is their site: http://www.puribahasa.net/ It's really amazing to be able to understand a bit what locals are saying, reply in bahasa, and talk to locals on remote islands where only few people speak English.
That's it for Jogja, next - Medan in Sumatra, the most horrible city so far :)