Here is a step-by-step process for analyzing a poem.
Draw lines through the poem to divide it into logical pieces. Look for the following to help you:
Stanza breaks
Syntax (where sentences start and end)
Rhyme scheme
Transition words like “But,” “So,” “And”
Changes in topic
Shifts in tone
Next to the poem, briefly write the main idea of each section. Consider the relationships between the different sections of the poem while doing this — how each main idea builds on, complicates, or otherwise relates to the previous one.
Make sure you can identify the poem's speaker and situation; these is crucial to understanding the poem's main ideas.
Write the poem’s theme. (This is the WHAT of what the poem is doing or showing.) When identifying theme, pay particular attention to the main idea of the final section of the poem — after the final shift. The poem's conclusion is usually the clearest statement of the theme.
Read through the poem one more time, making sure you understand how the main ideas of the poem are working together to create meaning and express the theme. Check that your main ideas are accurate. Check that everything makes sense given the speaker and situation you identified.
At this point, you should understand what the poem is saying and how the structure of the poem conveys that meaning. Now we can look for language that also helps convey that meaning. Annotate for literary devices that elucidate the main ideas of each section as well as the overall theme. You should be able to connect all the devices you identify to main ideas and theme. See this page for a reminder on the purpose of some basic literary devices.
See this page for an example analysis of a sonnet.