Be aware that a lot of modern poetry classes are terrible, and a lot of modern poetry is terrible. That doesn't exactly help find the good bits.
Cadence is about prosody, how all the sounds of a poem work together. It needs to be pronounceable, to sound good when read aloud or sung. This is usually not a concern in prose, unless it's written for oratory.
Most of what I write currently is narrative poetry, which tells a story; and free verse, which does not follow a set form. However, that's not all I write. I also love the hell out of form poetry and have written a lot of it. By a lot, consider that I'm currently working my way through an archive of 500 kigo, the season words used in haiku. I do rhymed, metered poetry with a particular fascination for repeating-interlocking forms. A quick introduction to my form poems:
I wrote short forms this January, so you can see a bunch of different ones linked in the report.
Haiku with the kigo:
A Conflagration of Dragons
This series has Six Races, each with its own style of poetry, plus the free verse.
Fiorenza the Wisewoman
Here you'll find some Italian forms and more free verse.
The Origami Mage
This series is mostly written in haiku or tanka verses.
Path of the Paladins
Several of these are written in quatrains (unrhymed, unmetered).
When I write lyrical poetry, it's often either ballad or couplet form. I happen to have posted a ballad just recently.
I have 2 poetry books out. Both have how-to materials attached that may interest you. In the top menu, mouse over My Works and then Books Written and then the title to bring up its extras.
It takes practice for most people to spot literary techniques and linguistic details. Good coaching helps. Few teachers have the linguistic aptitude to figure out how this stuff works, let alone explain it. Me, I put a couple of chapters on how to write poetry in my liturgy handbook because so little good instruction is available, and writing teachers copy them to use in classes.
Here is a list of poetic terms. Pick any one and search it like "poems with alliteration" or "how to write alliteration."
The Poets Garret is my go-to reference for forms online.
In paperback, I use The New Book of Forms by Lewis Turco
If you want to learn how to write poetry, I recommend Creating Poetry by John Drury.
Narrative verse uses a lot of subtle techniques that are used less often in prose, where you can't use too many or they get distracting. Among the ones I use frequently:
alliteration, assonance, consonance
poetic word choice (like kennings or colorful synonyms: compare "horse" and "steed")
simile and metaphor
allusions (compare with modern "eastereggs")
However, the line breaks have meaning too. They're breath points, little pauses. I typically use what are called end-stopped lines, meaning they break where a phrase breaks. Occasionally I use enjambment, breaking mid-phrase for emphasis. If you want to see why line breaks have meaning, read e.e. cummings.
There are also structures that aren't form but have patterns. Look at the ends of the lines; there's often a flow to them rather than big differences, although I wasn't as fussy about this in my earlier poems. Sometimes you'll see pairs of verses, or one idea spanned across a set of verses like 5 verses for 5 senses.
Some exercises to try, if you are curious about how to find the differences between free-verse poetry and prose:
1) Take a free-verse poem and similar sized piece of prose. Read both aloud. Is one easier to pronounce? Does one sound better?
2) Take a free-verse poem and a similar sized piece of prose. Highlight all the literary techniques you can identity. Which has more?
3) Delete the line breaks from a free-verse poem. Read it. Then read it with the line breaks. Does it feel or sound different? Compare the blocked version with prose too.
4) Take a piece of prose and put line breaks in it. Think about where you should put them and why. This is called a "found poem." Ideally you want to find a vivid piece of prose that sounds good when broken out in this manner. Try it with a nature magazine, they often have really nice descriptions somewhere.
5) When you're ready for more of a challenge, try analyzing a prose poem. Honestly, a lot of those are just dressy prose. I've written them but I'm not a fan. Some poets, however, seem to have a knack for the things.
Understand that you're not going to find the good bits of free verse by skimming. Print things out and look at them, look for patterns. Read them aloud, feel what your face is doing, listen to how they sound. A good poem has mouthfeel like good chocolate.
You know "Patty cake, patty cake, baker's man" ...? Suzette Haden Elgin pointed out that it has the unvoiced stopped consonants in order from from to back (P, T, K) in counterpoint to the vowels. That's what makes it fun to say and hear. Patterns. And they aren't just in rhymed, metered poetry; they're just easier to spot there. Skim and you'll miss them. There's a lot going on in a good free verse, but you have to look and listen for it.