How does each quatrain in Shakespearian Sonnet 20 connect to convey the meaning of the entire poem?

A quatrain is four lines long. There are three quatrains in a sonnet, followed by three lines at the very end. In a Shakespearean sonnet, the first three quatrains set up a question or problem, which is then resolved in some way by the final three lines of the poem.
"Sonnet 20" is part of what scholars call the Fair Youth cycle of Shakespeare's sonnets. The poem appears to be addressed to a young man with androgynous qualities.
The first quatrain gives the reader a first glimpse of the object of the speaker's devotion:

A woman’s face with nature’s own hand painted
Hast thou, the master-mistress of my passion;
A woman’s gentle heart, but not acquainted
With shifting change as is false women’s fashion;

Right away, we are told the youth's face is similar to a woman's, a fact which is emphasized by the speaker addressing the youth as "master-mistress." The speaker is not saying the beloved is entirely effeminate, but androgynous after all. The beloved's personality also contains feminine qualities, such as a woman's gentleness. However, the speaker also suggests the youth is superior to some women in that his gentle heart is not as fickle as the hearts of the worst of women ("but not acquainted / With shifting change as is false women's fashion").
The second quatrain intensifies the subject's infatuated fascination with the beauty of the beloved, emphasizing how he attracts both men and women to him through his gorgeous appearance:

An eye more bright than theirs, less false in rolling,
Gilding the object whereupon it gazeth;
A man in hue, all hues in his controlling,
Which steals men’s eyes and women’s souls amazeth.

Once more, the speaker shares his preference to the male beloved over women, claiming he is less shallow and more loyal than they. The emphasis upon the beloved's eyes and their vibrancy create an atmosphere both erotic and spiritual, since the eyes are often associated more with the soul than any other part of the body that the speaker could have referenced. Essentially, the beloved is gorgeous in form and soul.
The third quatrain has the speaker lamenting that the beloved is a man rather than a woman:

And for a woman wert thou first created,
Till nature as she wrought thee fell a-doting,
And by addition me of thee defeated
By adding one thing to my purpose nothing.

Here, the speaker complains that nature, by making the beloved man rather than female, thwarted the speaker's ability to be with the beloved in a romantic and/or sexual manner. These lines can be interpreted in quite a few ways. It is common to see the speaker as gay or bisexual, and that the last line of the third quatrain refers to society's refusal to allow the speaker and his beloved to be together. On the other hand, some argue the line suggests the speaker is not attracted to men generally since he complains of that "addition."
The final lines resolve the speaker's erotic complications by having him essentially give up on his beloved. The speaker is resigned to the idea that the beloved is meant to be with a woman. Since he cannot be with the beloved sexually, he will treasure the love they share and allow women to have use of the beloved's body.