What is reading and interpreting non-prose?
All literature is generally broken down into two categories: prose and poetry. If something is classified as non-prose, then what you are dealing with is poetry. So, your question is really about reading and interpreting poetry.
There is no one way to interpret poetry. It is a creative endeavor. For example, you can read poetry from a historical or sociological perspective. You can ask questions about the author and his or her times. What does the poetry say about the spirit of the age?
Since poetry uses many literary devices, you can also focus on these and ask what the literary qualities say about the poem. For example, does the alliteration or assonance say something about the meaning? Or does the meter and rhyme of the poem play a role?
Here is an example. "The Tyger" by William Blake is written in trochaic tetrameters. The feeling is one of a hammer hitting an anvil, which is also mentioned in the poem. This insight is a process of interpreting a poem.
To read non-prose, which usually consists of mainly sonnets, songs, and poetry is probably easiest to start with numbering the lines, that is, if they're not already lined with numbers. This way you can start with the first line and work your way down to the end of the sonnet or non-prose piece.
To begin to interpret non-prose, is an entirely unique process. Many of these non-prose selections for classes have already been interpreted, so there is usually a central theme of what most believe the meaning of the non-prose is trying to convey. However, you can check out the many scholars that have spent countless hours pondering the non-prose to form their interpretations and see what clicks for you to come up with your references.