If you want to know all about the beginning of the Holy Grail quest, go check out Malory, because this book's just not going there.
All we're going to get is that soon after Arthur had his bright idea, all of the knights of the Round Table set off to find the Holy Grail.
Two years pass, and the survivors of the quest start showing up. Half the knights are missing—presumed dead.
Gawain is the first to come back, and tells of a boatload of Sir Galahad's adventures, and how he's pretty full of himself.
As for Gawain, he had an adventure at the Castle of Maidens (which he doesn't want to talk about in front of the Queen), where he fights seven knights (along with Sir Uwaine and Sir Gareth—and Galahad also shows up).
Later, a hermit explains what this fight means. It's an allegory, so Gawain has basically fought the Seven Deadly Sins.
Turns out, only someone who has confessed and been absolved of his sins can even have a chance at finding the Holy Grail. Too bad for Gawain.
No adventures for Gawain. There's nothing out there for him.
Unfortunately, though, he accidentally kills Sir Uwaine, because he doesn't recognize him.
Gawain has also killed King Bademagus.
Arthur tells him to just go to bed and really think about what he's done.