One of the biggest things that distinguish poetry from other writing is how it uses form itself to create meaning. So when you start studying poetry, the key is to understand the variety of formal elements (meter, rhyme, lines, stanzas, etc.) and how they're used to create both formal verse and free verse. Once you can understand how poems are constructed, it's easier to discern any meaning they have.
Here are a couple books to help provide a primer on poem construction.
A Poet's Guide to Poetry by Mary Kinzie
This is geared toward how to write poetry. In addition to explanations about different formal elements and techniques, it also provides writing exercises and a small collection of poems themselves.
A Poetry Handbook by Mary Oliver
I haven't actually read this, but I've seen it recommended by numerous trustworthy sources.
In addition, I've found a good way to learn about poetic formal elements is to see how they developed over time, because most poetry is built off the poetry that came before it (even if in repudiation). So in conjunction with the above, I'd advise getting a comprehensive anthology organized chronologically. The Norton Anthology of Poetry has always been a go-to. These can be a bit pricey, but worth it. You could also search for older editions, too, since it's the history you're after and losing the last decade of poetry or so won't hurt you in this case.
I'd recommend getting all three of the above and start working your way through them.