To start this quest, talk to Duke Horacio who is on the second floor of Lumbridge Castle.
Duke Horacio will give you an air talisman which must be given to Seridor who is in the basement of the Wizards Tower. Then, Seridor will give you an item which you must give to Aubury (runes seller) in Varrock, he is in the backstreets. Then Aubury gives you another item to take to Seridor. Seridor will keep talking until it finally announces that you have completed the Rune Mysteries Quest.