The university of magic in Harmonia would just requiring you to finish the side quest in Notice Boards (The Goblins on the Road, The Vampire In Eldor and its consequent side quest that follows; Real Bad Vampires in Eldor and additional quest if you have Hakkon: Monster Hunt Quest. If you finish those. Gabriel will appear in Eldor inviting you to the Magic University. Assuming that you have payed your tuition first to Virgil (it is 1000 gold I think and it is required). It is also though not highly recommended to take your uniform from Flynn's after you paid the tuition.