Look at the form. Most poetry will be done either in a free form or in a specialized cultural form, such as a haiku or a sonnet. Knowing the difference between these will also give you a hint into understanding what the poem is about. You can tell the form by looking at where the rhymes are or aren't, as well as how long the poem is. For example, a haiku will only be three lines long, a sonnet will be 14 lines long and free form will take up as much room as it wants without any rhymes.
Read the poem through a couple of times. If you are going to analyze a poem, you don't have the luxury of breezing through the words and turning the page. A good poem will take a couple of times to read through before it makes sense.
Look into what the author is trying to talk about. It's time to start breaking down the poem to get some sort of meaning out of it. You can easily do this by starting at the end of the poem and comparing it to the beginning. Most likely, the end thought will be a turning point of what the author realizes out of the theme that has been written about. The introduction is the idea or thematic material that the author will be working with throughout the poem.
Find the breaking point. Towards the middle or end of a poem is a turnaround point. This is the point where the story, ideas, thoughts or descriptions will change in order to get to the point at the end of the poem. For sonnets, this will be in the last two lines. For haiku, the middle line is always the turning point. For free form, you can find the turning point anywhere from the middle to the end.
Break it down even more. Once you know the breaking point, you can find how the poem shifts from one point to the next. Start doing this by finding the major thought in each of the stanzas or paragraphs. From this, look even closer at the literary techniques being used. Find the metaphors and similes that compare the ideas to a thing. Look for personification that changes an object into a person. Keep an eye out for the different types of descriptions in each line and how this relates to the end of the poem, the beginning and the turning point.
Put it back together. By this time, you should have an idea of the individual lines and what they mean, as well as the different sections of the poem. From here, you can put the form back together and truly understand what the author is trying to say, from the beginning to the end of the poem.