How do I properly analyse a poem?

Firstly, before approaching the poem, note down the form of the poem, including the number of stanzas, the number of lines in the stanza and the rhyme scheme (if there is one). It is then important to re-read the poem a couple of times to gain the underlying meaning so that you understand what the literal devices are used for.

Then, underline or circle any words that stand out to you visually, rather than for their meaning. This could include repetition, longer words surrounded by shorter ones, alliteration, assonance, sibilance and so on. Once you have identified these literal devices, then use the meaning of the poem to demonstrate how these devices enhance what the poet is trying to illustrate.

Knowing the meaning of the poem, make sure to have regard to metaphors and similes (and to not confuse the two), as well as pathetic fallacy, if the environment is reflecting the mood, which you are now aware of.