Poetic analysis covers all devices from elements, like structure and rhyme, to techniques, like metonymy and metaphor. In this present format, I can provide a rudimentary analysis that you can expand upon.
The structure of Sonnet 1 keeps the Petrarachan form in that there is one volta at line 9 but follows the English sonnet form, immortalized by Shakespeare, in tht it has 3 quatrains (4 lines) and an ending couplet (2 lines). The rhyme scheme is ababababcdcd ee. The first two quatrains repeat the abab rhyme but the third quatrain turns to a cdcd scheme that reinforces the volta, which is a turn in subject under the central topic of the poem. This line 9 volta turns from talking about what Astrophil sought to do:
Loving in truth, and fain in verse my love to show,
That she (dear She) might take some pleasure of my pain:
I sought fit words to paint the blackest face of woe,
Studying inventions fine, her wits to entertain:
and turns to how he failed to do it:
But words came halting forth, wanting Invention's stay,
Invention, Nature's child, fled step-dame Study's blows,
The paradox or problem of the sonnet is resolved in the couplet when Astrophil tells what was revealed to him: "Fool," said my Muse to me, "look in thy heart and write."
The most prominent poetic technique in Sonnet 1 is Sydney's use of metaphor to illuminate the nature of poetry. Poetry is painting, "paint the blackest face ...." Poetry is "inventions fine," fine words in newly created expressions. Poetry is "fresh and fruitful showers" of imagery and imagination. Finally, poetry unwritten, like Astrophil's, is a full-term but undelivered pregnancy and the "throes" of labor pains that precede birth: "Thus, great with child to speak, and helpless in my throes, ...."