How do I start a thorough analysis of the poem "The Good-Morrow" by John Donne?

To write a solid analysis of this poem, you might begin with the significance of the title "The Good-Morrow."  Take this title, which means "good morning," and apply it to the structure and themes of the poem itself. Notice that in the beginning of the poem, the speaker contrasts the lovers' lives before they met each other and after. To emphasize this contrast, he refers to sleeping and waking. Before the lovers met, it was as if they lived in the "Seven Sleepers' den" or a dream world.  

The second stanza takes this contrast a bit further. The title of the poem "The Good-Morrow" is mentioned here as the speaker goes on to show how the lovers are presently. It's as if their souls have awakened to the knowledge that their love is all they need, that their love forms a world which makes them superior to all still searching for fulfillment:

Let sea-discoverers to new worlds have gone,

Let maps to other, worlds on worlds have shown,

Let us possess one world, each hath one, and is one. 

In the last stanza, the speaker describes the physical position of the two lovers, who are now wide awake and looking into each other's eyes. In their eyes, they see a world without coldness or harshness ("sharp north") and without end ("declining west"). Using the metaphor of eyes which reflect the images of each other, Donne writes,

My face in thine eye, thine in mine appears

and then connecting the roundness of the eye to the globe or world, the speaker shows both the perfect union and the eternity of their love. The lovers are saying good morning to each other, the union of their souls, and the everlasting quality of their love.

If our two loves be one, or, thou and I

Love so alike, that none do slacken, none can die.  

Aside from the title and its connection to the contrasts and metaphors within the poem, you might look at how the sounds of the worlds reflect the poet's meanings. For instance, the last three lines of the first stanza contain repetition of the long "e" sound. This assonance creates a gentle and reverent tone as the speaker describes his lover's beauty and her superiority to any other woman he has ever met. The internal rhyme in the last stanza, 

My face in thine eye, thine in mine appears,

brilliantly echoes the idea of unity: "thine" and "mine."  

There are many other facets of this poem that you might explore. I have included some links below that may give you more ideas.