The Wart started talking before he was half-way over the drawbridge. "Look who I have brought," he said. "Look! I have been on a Quest! I was shot at with three arrows. They had black and yellow stripes. The owl is called Archimedes. I saw King Pellinore. This is my tutor, Merlyn. I went on a Quest for him. He was after the Questing Beast. I mean King Pellinore. It was terrible in the forest. Merlyn made the plates wash up. Hallo, Hob. Look, we have got Cully."
The quote that tells all about Wart's Quest is in Chapter 4 of Book 1 of White's tale. Excerpted above, you can see the quote tells all about Wart's quest adventure. Cully had gotten separated from Wart and Kay because of some mismanaged falconry on Kay's part. Wart sacrificed his safety to stay near Cully to try to coax him back. Part of Wart's motivation was his instinct for falconry; part was his affection and concern for Cully; and part was his respect for Hob.
The effort to rescue Cully from the consequences of his flight away from Wart and Kay led to far more adventures and near mishaps than Wart could have imagined. As he says, he was shot at; he encountered King Pellinore; he met Merlyn and gained a tutor; he reunited with Cully; he made other new friends; he saw breakfast plates, talking and laughing like children, scurry off to wash themselves up. His Quest, as he puts it, was to find his tutor, Merlyn, though it started out as a rescue mission for Cully.
"This is my tutor, Merlyn. I went on a Quest for him. ... Hallo, Hob. Look, we have got Cully."