north
Make sure to factor in the astronomical $20 US entry fee for foreign tourists at Borobudur and $18 US at Prambanan before you show up. They of course accept the Indonesian equivalent. Free tea/coffe and bottled water included.
The massive Buddhist temple of Borobudur, the largest Buddhist monument in the world, is 40 minutes away by car and one of the main drawcards for visitors to Yogyakarta. Many tour buses minivans leave Yogyakarta at 5AM for an early visit to Borobudur. This is a convenient way to skip the crowds which arrive mid morning, although a little more expensive between Rp 45,000-50,000 including a simple breakfast. However, the extra cost could save you money if tour participants chip in for a tour guide.
Kaliurang is a small town on the southern slope of Mount Merapi. One reason to visit this town is the Ullen Sentalu Museum (http://www.ullensentalu.com/), an exotic museum dedicated to the culture and life of Javanese royalty. The museum is built to the surrounding landscapes, with gardens, sculptures and an overall tranquil and natural environment. There is also a colonial-themed restaurant located in the grounds called Beukenhof.
Ketep, a pass located in the slope between Mount Merapi and Mount Merbabu offers a great vantage point for a spectacular view of both mountains. This site has a small cinema which regularly shows an interesting documentary of Mount Merapi. Located in Muntilan, east of Borobudur, 40 minutes from Yogyakarta by car.
It can work out quite economical if you rent a scooter for 50,000 a day and drive it to Borobudur and Prambanan instead of taking the tours. You'll need another 10-15,000 worth of petrol though. The roads to both sites are pretty wide and in good condition. If you leave early in the morning you can make it to Borobudur in a little under 60 mins. It's really quick and even cheaper if there are 2 people instead of one.