How do I analyse a poem?

Firstly, select a quotation that either has imagery (e.g. a description of flowers), interesting punctuation (e.g. a pause in the middle of a line), symbolism (e.g. red could mean love or anger) or alliteration (e.g the red rose). For my example I've selected "Parking at hospitals. Holding hands under / Veteran magazines". This line has a) alliteration - the same sound 'h' at the beginning of 3 consecutive words, b) a pause (the full stop) and c) the imagery of the couple holding hands. Secondly, apply meaning to these characteristics. 'H' alliteration creates a breathless feeling which would connect to crying and the grief of the poem. The pause creates a contrast between the scary hospital and the holding of hands - emphasising the need for comfort. The poet makes clear that companionship and comfort are needed, so in the wider context of this poem this suggests that everyone's role is significant.
To improve your analysis I would suggest repeating this technique until the meaning behind the techniques becomes easier to find. Some ideas can be learnt e.g. 's' alliteration is normally calming whereas 'b' is aggressive, however many rely on the context of the poem as a whole - so make sure you understand the 'message' of the poem first. My example was about grief so these techniques are linked to that. If you struggle with the poem itself I suggest practising the analysis on your favourite song first - a more familiar piece of text. Remember that poems are about the 'tools' a poet uses to convey a message, analysis is understanding that message.