In my experience, there are 3 ways to read a poem:
1) SURFACE: This is what you get out of reading it once, considering it a few moments, noting ideas and feelings that 'surface,' then moving on to the next. You either like it or you don't. It either grabs you or it doesn't. If it doesn't you don't go deeper.
2) DEEPER: You've read it many times on different occasions (meaning: you keep coming back to it over a period of time). You've begun to notice its elements separately--sound, rhythm, figurative language, multiple meanings of words, allusions to myth, other writers, other poems, history, literature, politics, religion, etc. and how these moving parts work together to illuminate and support (or sometimes challenge or reject) the ideas contained in the poem. They inspire you to write and you might try to emulate the poet's style.
3) CRITICAL: I mean this in the best sense of the word. You've read, 'deeply,' the poet's work, and have a sense of his/her voice and interests. You're able to connect, compare, and contrast one poem with his/her other poems (or one book with other books of his/hers) and see a development in thoughts, ideas, beliefs, technique, and skill and how these relate to the world of poetry (past and present) as a whole.
Poets that attract many critical readers define contemporary poetry. They choose poetry's next frontier and head out to conquer it. You might guess where he/she is going next and attempt to meet them there!
With some books, you fall straight into a deep reading. With others (unfortunately), you skim superficially. Whether or not you get to the 'critical' stage with your readers, depends on you and how dedicated you are to the craft of poetry.
To go deep with your readers, you have to go deep with yourself first.