How do I analyse punctuation and sentence structure in a poem?

First of all, my advice would be to come to punctuation last. Poets use punctuation to shape the flow of the text, so coming to it too soon won't actually provide much insight into the poet's intentions. Instead, first identify any literary techniques the poet uses and what the techniques tell you. Then you can look at how the poet uses punctuation to split up the text. Do they use a series of commas to create a long list of frantic thought? Or do they use a short simple sentence to isolate an important idea? It's important to always think of how poetry sounds when it's spoken out loud: would the punctuation make a speaker reading the poem sound unsure or blunt? Excitable or thoughtful? How does the punctuation ADD to what the words tell you?