Rather like music theory, having a highly analytical mind is a boon...eventually. It's so easy to dig into the mathematical structure of it all and not deal with the actual thing. So start with the rollicking and absurd, and
read it aloud!. Lewis Carroll's Jabberwocky, for example. I have yet to meet someone that doesn't get a grin out of that. Isherwood, 'The Common Cormorant'. These are poems that beat you over the head with their structure and meter. Limericks, doggerel, songs from Shakespare plays...and pay attention to where it seems to stutter or skip, where it doesn't work. That is the beginning of training your ear, and recovering the ability to feel and enjoy the sound, making that part of the reading experience. Technical prose has trained that out of us as a matter of self defense because so much technical prose is only tolerable with your ear firmly swathed in cotton wool and stored carefully away.
When you get past the level of limericks and children's poems level, suddenly you have interpretive choices to make because the precise flow and timing of sounds can work different ways and some work better than others, and as a reader you have to bring an interpretive faculty to these works. When you get to something like John Donne's poetry, you're now deep in "every bit of timing and inflection matters."
For children's poetry to get started, Belloc's 'The Bad Child's Book of Beasts' or the Looking Glass Book of Verse are high quality collections. Then it's worth getting something like a Norton Anthology of Poetry as a way of reading very widely very quickly to find out what appeals to you right now, and explore that further.
Many people recommend the Oxford Book of English Verse. I don't, because I think Quiller-Couch had insipid taste and his editing ranges from the uninspired to the positively atrocious (what he did to John Donne's 'The Ecstasy' is horrid). I have found Garrison Keillor's 'Good Poems' to be particularly approachable, high quality collection of verse. I also suggest Ezra Pound's 'ABCs of Reading', though you shouldn't take what he writes about Chinese as anything other than a metaphor for his topic at hand.
Spend time with Shakespeare, of course. I recommend the Oxford complete works. I do not recommend the Yale complete works. Did my wife and I have a tiff about that? Yes, we did. Was I able to demonstrate by reference to passages that I was right? Yes. Do we only have the Oxford in the house now? Yes.
Robert Frost's complete works are cheaply had and a necessity for someone studying modern poetry. Two or three poems of his are butchered in every school class, and the full range of what he worked on is generally ignored, because it can require a very finely tuned ear to hear and interpret the force that his longer, narrative poems can produce.
But really, once your ear starts to come together, dig through the Norton anthology and use it as pointers to further things to read.