Gawain's quest is to fulfill the terms of the challenge he accepted from the Green Knight. At the start of the poem, the Green Knight visit's Arthur's court and challenges Arthur to deal him a blow, with the understanding that in a year, Arthur will travel to the Green Knight's castle and accept a similar blow.
Gawain is good Knight in the sense that he—mostly—does what is expected of knights. I think a larger question the poem poses for modern readers is whether these expectations are reasonable. Gawain is admirable in his acceptance of the Green Knight's challenge on behalf of Arthur, and in his honoring his pledge to seek out the Green Knight at the Green Chapel a year after he deals him what ought to have been a killing blow. His virtue is without question, but also suggests that he is a bit naive.
His encounter with with Lord Bertilak and his lady tests his virtue. Lady Bertilak's attempts to seduce Gawain are not successful, and in fact Gawain lives up to a promise to share with Bertilak anything he may have gained during the day, giving him kisses each evening. In fact, Gawain is hard-pressed to avoid betraying his host and offending Lady Bertilak. But he cannot resist the temptation of accepting her favor (a piece of green lace) which she says will protect him from harm in battle. Knowing that he must soon face the Green Knight, he accepts the token.
It is a matter of interpretation if we understand Gawain's weakness at this moment as a sign of his unworthiness as a knight. In the final lines of the poem, Gawain certainly feels that the lace is an emblem of his cowardice. Yet Arthur makes merry of this sentiment, and bids everyone at court to wear green in support of Gawain, and this suggests that Gawain's scruples are too great, or that, for Arthur, the whole adventure has been a kind of joke—one which Gawain did not get. Ultimately, it is Gawain's failure to live up to his self-imposed standards that humanizes him.