Hello, and welcome to Avaria!
To briefly answer your questions: Children are typically sent to a kind of elementary school called a maktab, though wealthier families sometimes hire private tutors instead. The usual age for going to a university is eighteen, though students may be accepted at a younger age if they show great promise in their earlier education. Someone aspiring to be a physician would spend at least part of their higher education at a bimaristan (hospital). Major bimaristans generally have an educational program (which may stand on its own, or may operate in partnership with a local madrasa), and students have a curriculum of classes while also learning hands-on at the hospital itself. This is usually under the guidance of one or more physician mentors.
For a little more information, you can check this out too: http://songofavaria.com:8080/mediawiki/index.php?title=Sirdabi_Caliphate#Education
Hope that helps! If you have more questions just ask.