I think that there are many ways to approach this. If your teacher or instructor or course has a way that has already been outlined, I think that you might be best set in following this format and then perhaps integrating any advice you get here would be a good plan. I have always felt that a good starting point would be to number the lines in a poem. This helps to give a frame of reference where some level of specific discussion can begin. I would then read it over, identifying what you think are important words or images that are constructed. The language used in a poem can help out tremendously in fully understanding meaning. Additionally, I think that it can be extremely powerful to fully understand the mental pictures or imagery that are present in a poem. In the identification of essential words, some examination as to meaning might have to be present in order to fully understand what is being argued. I think that at this point, some discussion about the literal and symbolic meaning of the poem might be worthy of examination. The literal meaning refers to what is happening on the page, in the poem. What is the exact story? At the same time, I think that the next level of discussion comes out of the symbolic meaning of the poem. What might the poem be trying to say about what it means to be human or to be alive? Usually, poetry seeks to broaden the individual experience into a universal one and it is here where the symbolic meaning is present. These are some of the tips that can be used to analyze a poem.
There are several ways to take a look at a poem.
First, read it in it's entirety to get a feel for the topic. Some poems topics are hard to decipher. Others are blatantly obvious.
Next, take a look at it's mood and tone. Try to determine if it is positive or negative, and beyond that what the specific feeling is about the poem. Is it happy, sad, apathetic, disgusting, or disconcerting?
Next, identify literary elements that you know. Are there similes, metaphors, evidences of personification, or puns? Are there allusions, or poetic elements that have significance in terms of meaning.
Next, look at form. For some reason, every once in a while a poem is set up according to its meter and rhyme and these features contribute to meaning. In a sonnet, often the first 8 lines centralize on one topic while the next 6 take it somewhere else.
Finally, look for a shift in the theme. In a sonnet for example, these 8 top lines stay on a certain issue, and before you know it the next line feels like another stanza all together. It isn't. It is a shift.
You should also look for literal and figurative meanings as you suggest.
Good luck with poetry!