We went to the Hindu site, Prambanan, first, as it's closer to the city of Yogyakarta (around a 30-minute ride, depending on traffic: which as in so many developing countries is heinous thanks to a lack of infrastructure investment). We rented a car and driver for the day (which cost around $15? I have a terrible memory for numbers). You can also take the bus, but it's more pleasant to have your own air-conditioned transportation when traveling with a toddler.
Because Prambanan is a UNESCO site, it's very well maintained and has all the facilities you might want: a restaurant, a museum with English signage, lots and lots of vendors trying to sell you souvenirs, water bottles, umbrellas, guide books, etc, etc. The entrance fee is pretty steep for Indonesia ($18 for foreigners; Indonesians pay much less) but since a visit can easily last the entire day I thought it was a good value. You also get free water and coffee with the ticket!
Ami a kiss.