If you are french or looking to study the language in the city, IFI-Lip it’s a great place to start.
Located in the center of the city, the institute it’s ran by french people for french and non french people.
What can you find there? Private or group classes, DELF exams every now and then and a great, truly great and probably most complete french library in the city. With just 5 minutes paperwork and a small fee of 150K a year (If im not mistaken!) you can get as many books as you want (But just one at a time).
Check their website: http://www.ifi-id.com/yogyakarta