How could I get more out of reading a poem?

Well, of course you will have been told about the various style devices of poetry, and what to look out for in studying a poem, but it is true that sometimes breaking a poem down into its component parts can lessen its appeal. It is important to maintain the initial reaction you had of reading that poem, the feeling it gave you, onto which you can then analyse the form, rhythm, rhyme scheme, content, devices such as alliteration etc. The combination of devices which form the poem as a piece of work should then contribute to an appreciation both of it as an artistic piece of craftsmanship, and as a piece to which you can respond.

The role of the reader itself is an interesting one in literature, as the Romantics emphasised that more personal, subjective, reaction, and then as time passed various movements placed emphasis on the work itself, viewing the reaction it gave the reader as secondary to its aims. Yet, the form of the question you asked me suggests a subjective, personal, emphasis, and so I would recommend an close analysis of its components as, hopefully, complementing and adding to your appreciation.