In late elementary-early middle school I read this series I got from the scholastic reader pamphlet. I can't remember what it was titled to save my life. It may have had "chronicles" in the title. It read kind of like a generic Harry Potter. It had a magical school that was a castle, but I think it also involved Knight training and quests. I loved it at the time.
While it doesn't have "Chronicles" in the title, The Circle of Magic series by Debra Doyle and James D. Macdonald fits plotwise. In the first book, School of Wizardry, we encounter Randal, a squire in his Uncle's household, training to eventually become a knight, who instead goes with the wizard Madoc to learn magic at a school which, yes, is in a castle. The book was first published by Troll Communications (a company that Scholastic acquired), and later, Scholastic published a copy of it.