Start by reading it aloud (or in your head if in an exam) to get an idea of the rhythm and rhyme. There is a lot to say about both: rhyming patterns say a lot about the poem (e.g. ABBA/ ABAB tends to have a structured, regular form, and rhyming syllables together has a more comical effect: 'willfully' and 'skillfully'.
Listen to the patterns in the poem: is there any sibilance ('s' sounds)? Or alliteration? Or similes and metaphors?
And is the register formal or informal? (i.e. is any slang or informal language used or are the words and phrases quite intellectual.
Finally, look at if it's first, second or third person. That can also tell you a lot about what the message is/ what the purpose of the poem is.
Read it aloud as many times as necessary- poetry is made to be read and the best way to understand it is to hear it.