Thoughts from an amateur.
I’ve written before on how to to write a poem. Followed by how to truly write a poem – study Mary Oliver’s A Poetry Handbook and then practice.
But reading a poem is a whole different pack of monkeys.
I developed this weird method to help me absorb the poems I read. It slows me down, so I don’t rocket through the lines. The aim is to bury the verses in my subconscious.
See if it works for you.
First I read the poem to myself. From the first verse to the last, all the way through.
Then I’ll read the poem from the end to the beginning. I read line by line, from the final verse, back up to the opener:

Reading it backwards is like reverse engineering. It helps me see the poem’s structure. How each verse builds up to the final one.
After that, I’ll read the poem beginning to end again, but this time out loud.
Reading out loud helps you find the poem’s rhythm. I’m sure there’s things like meter and tone involved as well, but I won’t pretend to know how.
Then I’ll read the poem in reverse order again. But this time in full blocks. Starting from the bottom of the poem to the top:

While reading I’ll keep a pencil close. If the poem rhymes I search for the rhyming pattern by underlining all the rhyming words.

Once finished, I’ll log the date, author, and name of the poem in my steno book. Keeping a record gives me a sense of progress.
It’s a practice I stole the from director Steven Soderbergh who publishes a yearly log of what he’s watched, read, and listened to, on his site.
This how I read a poem. You may read a poem once and bin it. And that works too.