First, simply read a collection of Petrarch's sonnets. Take notes about the poems that strike you as especially powerful, marking down any images or feeling or phrases that stand out. Notice how most of the poems are descriptions of Laura, his idealized muse. Most of Petrarch's poems are about the suffering he must endure while he tries to make Laura love him. Try to determine if the sonnets tell a story about the relationship. Do they end up happy or is there a tragic ending?
Note the form of the sonnet. The number of lines, the number of syllables per line, and the rhyme scheme are what makes these poems Petrarchan sonnets.
Petrarch's sonnets are 14 lines. They are broken into two sections, an octave of 8 lines and a sestet of 6 lines. Usually, there is a turn in the ninth line which indicates a move from a problem (described in the octave) to a resolution (described in the sestet).
The rhyme scheme reads "a b b a a b b a" for the octave and either "c d e c d e or "c d e d c e" for the sestet. Each letter stands for a sound. The first four lines, for example, might end in the rhyme sounds "me / door / poor / thee."
Choose whatever moved you most and start brainstorming or writing about that theme, image, or sound. Don't worry yet if what you write fits the form of the sonnet. You are just getting ideas.
Once you have many ideas, try to write them in the form of the sonnet. Decide first how to group your ideas. Remember that Petrarch always used an octave to develop a problem and the sestet to present a solution. Do you have a clear problem and resolution?
Look for rhymes among your ideas. What words do you like best? What similar words rhyme with those words?
Once you have a full draft, read it aloud. Listen to what sounds good and what may need revision.
Revise your poem. This part often takes the longest. Some poets even say it takes them years to completely revise their poems.