First thing first, it is important to recognize that this is from a collection of sonnets from Astrophel and Stella and that this particular sonnet addresses the moon (or the moon goddess, Diana or Artemis). Astrophel would be a word signifying "lover of the star" and Stella "the star", so we know it's a love poem. Having the poem address Diana (or Artemis in Greek) shows that the speaker feels that the his audience is permanently out of his reach: Diana is known for her coldness and her virginity, and her refusal of contact with men. The moon therefore serves as a kind of poetic foil for his love (who he cannot reach) and a cold, unknown, and forbidding place that will not be touched by men, far away, and undoubtedly feminine in nature. So that's where you start. I'll go by couplets, to make this easier to paraphrase for your purposes.
With how sad steps, O Moon, thou climb'st the skies !
How silently, and with how wan a face !
The poet is noting in the first lines that the Moon rises in the sky in what he sees as a sad slow progression, in silence, and is pale-faced- which is just an observation of the pale surface of the moon as it rises.
What, may it be that even in heavenly place
That busy archer his sharp arrows tries?
The poet here questions why the moon might be sad. Could it be the same reason that the poet is sad; that the archer (Cupid) has shot her with his arrows, and she is in love like he is, but unable (or unwilling) to be with the one that is loved?
Sure, if that long-with-love-acquainted eyes
Can judge of love, thou feel'st a lover's case;
The poet here notes that the moon has always looked down upon all lovers from her perch in the sky, so she must be able able to recognize love with her own eyes, and judge of it, so she must understand what it means to him to love, and how he feels (his "case").
I read it in thy looks; thy languisht grace
To me that feel the like, thy state descries.
The poet is saying here that he can see it in her slow pace and pale face that she must feel it too, and know it from experience, and he can know how she feels, because he feels it the same way.
Then, even of fellowship, O Moon, tell me,
Is constant love deemed there but want of wit?
Here he is saying, since you and I are "fellows" in love, please tell me, from your perspective, am I (the poet or the narrator) out of my mind/lacking in insight for being so steady in my love, when love is so temporary and is never returned?
Are beauties there as proud as here they be?
Do they above love to be loved, and yet
Those lovers scorn whom that love doth possess?
Do they call virtue there, ungratefulness?
I'll take the last two couplets together since they have an enjambement. Here the poet asks if all the subjects of love (beauties) are proud and resistant to being loved in the heavens as they are on earth. And do those beauties love to be loved, but also look down upon (like the Moon does!) those who love them, never returning, and spurning that love, but simply adoring the admiration of their admirers? And what can make someone so ungrateful that they would spurn true constant love like that, unless they thought ungratefulness was a virtue? The last couplet is always a twist on the rest of the sonnet- in this case, it's a way for the poet to call the Moon (his distant and ungrateful love) virtuous, but only if earth and sky were somehow, and impossibly, flipped.