I spent a month traveling aroung Indonesia last year... And boy it was without a doubt one of the highlight of my trip. If you like traveling, meeting people, nature and culture, you're going to have a blast dude ! The only place a bit sketchy is the very north of Sumatra, around Banda Aceh.
As for Bali, like it's been said above, Kuta is fun for a night or 2. We can say what we want, we're young and we usually like to party :) . But your should definetly check Ubud (stay there at least a few night), and navigate around the island from there. Last time I rent a car, but some motorcycle are fine too, and just go get lost :) .
One thing, you should absolutly check the coast road that goes from Amlapura to Amed... It's breathtaking. Find a guesthouse around amed and spend a night there, you won't regrett it.
And now Java, wich is the good part, there is so much to see there. Gunung Bromo is nice, the view is exceptional but it's full of tourist. Beside that, you have to stay for a few night in Jogyakarta, it's the the heart of the whole country. Borobudur and Prambanan are two of the world most amazing temples, be sure to check them out. The train ride between Jakarta and Bandung is a must aswell, it last 2hours if I remember right, and you'll get a grasp of some of the prettiest indonesian sights. And on top of that, Bandung is a pretty nice place, student town in the mountains, I liked it.
But boy, if there is one thing you shouldn't miss in Java, it's the Dieng Plateau. You can take a train to Purwokerto, find a local bus or something, and travel to Wonosobo. There if it's not already night time, take the last bus to Dieng and find a ghuesthouse there (there are 2). If it's night time, sleep one night in Wonosobo, the road to dieng will be close, you'll understand why when you'll take it :p. Spent the rest 2 3 days exploring the platau, its the most amazing place in Java. It's cut from the world, at like 2000meters high, with volcanoes, rice terrasses, the oldest temple of Java and so on. This is the real Indonesia.
There you go, hope that helps !