It can certainly be difficult to analyze a poem, but the strategies previously suggested are certainly helpful. One problem some of us have is that we get nervous when someone says "poetry." So we should talk about what poetry really is.
A poem is big idea or a big picture in a tiny package. The poet selects his or her words and ideas very carefully, because he or she does not want to use as many words as a person writing a story or novel would write. This means that when we read a poem, we should realize that probably every word is very important, that every word has a reason. If you have ever used Twitter, or read Twitter messages, you are seeing exactly the same idea in action. The writer cannot use too many words but must convey a big picture or big idea in that small space. So, in Twitter, as in poetry, every word counts!
One good strategy described in the previous response is to try to understand the meaning of the lines you are reading. Sometimes the best way to do this is to take each line or a few lines and write next to them what the meaning is in your words. That is often particularly useful with older poems written in styles you are not accustomed to reading. Once you paraphrase, which is what you are doing, you can focus on the surface meaning first.
Another aspect of poetry that you will want to focus on is imagery. Imagery is just a way of talking about the "word picture" the poet is trying to paint for you. Of course, there is more than a picture in many poems. A poet might want you to experience a sight, a sound, a smell, a taste, or a touch. The words the poet chooses to help you experience these sensations are called imagery.
If you are able to "translate" the surface meaning of the poem and understand the "picture" the poet is trying to paint, then you can start to think about deeper meanings. Ask yourself some questions about what the poet wants you to see and feel. At that point, you are well on your way to becoming a good analyst of poetry. Good luck!
In general, studying poems is not entirely different than reading any particular text. I think that there will be different approaches identified to you, but one thing that helps me is to use what I know about the author. Sometimes, understanding a brief biography of the poet would help in being able to analyze meaning and messages of a particular poem. The author's experiences, beliefs, upbringing can all play vital roles in being able to decipher the meaning of the poem.
In terms of analyzing themes in a poem, I think you want to ask yourself the question,"What is the poem trying to tell me, the reader?" The poem's theme is its message, the purpose of its being. Each poem's theme is a statement of belief and wants you, the reader to accept or "buy into" this idea. When you are reading a poem, ask yourself what is the literal meaning of the poem. This is going to have you examine the poem's surface understanding. Then, you will be able to approach its overall theme, or message that it is trying to convey to you. Being able to analyze a poem on both the surface meaning and its thematic purpose will be able to assist you in analyzing the themes of certain poems.