Yogyakarta is home to two very famous temples: Borobudur and Prambanan. Both are world heritage sites and popular tourist destinations in Yogyakarta. They are ~1 hour outside the city in opposite directions which can make for a long day if you decide to do both. Most people divide them into 2 days.
Borobudur is a 9th century Buddhist temple and one of the greatest Buddhist monuments in the world. Tourists can see the sunrise from the top of the temple but they must enter through Manohara Hotel and pay the extra fee. Piece of advice: pay it.
An hour away from Borobudur is Prambanan, the largest Hindu temple complex in the world. The grounds also contain a couple smaller Buddhist temples that are currently being restored. Unlike Borobudur, the area is more commercialized. Lacking the beautiful and simplistic landscaping of Borobudur, Prambanan feels a bit like an amusement park. Walking paths are wide enough for multiple golf carts and tacky souvenir shops line the streets. Not to mention, the exit is strategically placed; you must walk through a large food & souvenir market to leave. To be honest I was a little disappointed by Prambanan, especially having already seen Borobudur. Don’t get me wrong the temples were beautiful but the grounds lacked the “feel” of a world heritage site.