Come Sleep! O Sleep, the certain knot of peace, The baiting place of wit, the balm of woe, The poor man's wealth, the prisoner's release, (1-3)
The speaker is tired. That's a kind of suffering if we ever saw one. The lines talk about the ways in which Sleep can alleviate different kinds of suffering (woe, poverty) and thus foreshadow the speaker's own complaints later in the poem.
With shield of proof, shield me from out the prease Of those fierce darts Despair at me doth throw; (5-6)
As if we didn't know that Despair was causing some serious suffering, the speaker rhymes "throw" with "woe" (2) and "low" (4). This sequence of rhymes emphasizes at the level of sound what the speaker tells us: he's in a state of woe and feeling low (hey look, we rhymed internally).
O make in me those civil wars to cease; I will good tribute pay, if thou do so (7-8)
Some of the speaker's sufferings come from the inside. There's some really bad stuff going in inside of him. Civil wars? It sounds like a metaphor for some conflicting emotions that are really doing a number. He wouldn't call them wars if they weren't pretty bad.
Take thou of me smooth pillows, sweetest bed, A chamber deaf to noise and blind to light, A rosy garland and a weary head: (9-11)
The speaker offers Sleep his "weary head." This is a very creative way of asking Sleep to relieve him of his suffering. Sure, we think for a second that he's taking off his head, but he really means it more figuratively: take this weary head and put it to sleep.
And if these things, as being thine by right, Move not thy heavy grace, thou shalt in me, Livelier than elsewhere, Stella's image see (12-14)
The word "heavy" is interesting. The whole poem has been about how Sleep is a balm—like, um, Chap Stick. "Heavy" isn't always a good word, though. It can be associated with sadness, for example, as in "It is with a heavy heart that I give you an F." Maybe Sleep is a cause of suffering too.