This is a good question. There is no one way to analyze a song. So, there will be many different opinions. In light of this, let me give you some suggestions.
First, you can analyze a song in terms of its lyrics. If you go this route, then treat the song as poetry. You might consider the artistry of the words. Consider things like word order, rhyme, imagery, allusions, and the like. Closely looking at the words can be a very helpful and insightful.
Second, you can analyze the song in terms of it music. You will need to be someone who knows something about music, but this can be very fruitful as well. You might want to consider things like how the notes work together, the effectiveness of the crescendos and the like.
Finally, you might want to examine things like how the song make a person feel. This is, of course, subjective, but it can be helpful.