First and easiest step - find yourself a good poetry compilation and use it frequently for read aloud time. I personally love A Child's Anthology of Poetry, which my husband bought several years ago as a non-picture book read aloud that he could easily read to the kids in short chunks. It has become an invaluable resource for us and really lots of fun for adult reading. The poems are not "children's poetry" in the sense that they were written with children as the primary audience, but were chosen as poems that will resonate with children but also provide a strong baseline familiarity with the major body of poetic literature. Its a broad and diverse collection, appropriate for a range of developmental levels. I most likely wouldn't read "The Raven" to a five year old for example, but the variety certainly gives a nice scope to the collection. On that note, my husband once read "The Rime of the Ancient Mariner" to my oldest daughter at 5 years old, and while I was horrified that she was going to have nightmares, she actually did enjoy it and seemed to comprehend a lot more than I would have expected. Treat poetry reading as fun reading - please don't belabor the process by trying to unpack heavy meaning from the poems. Of course it can be fun to wonder out loud what happened in the poem, or are there times when you feel like that poem describes? Just don't turn what should be light and fun into a lecture at this point. A few other excellent compilations: Favorite Poems of Childhood and The 20th Century Children's Poetry Treasury.
Oh copy work, you are my favorite "kill two birds with one stone" trick - or in this case three or four birds. Copy work improves handwriting, spelling, and vocabulary, while also giving children the opportunity to slow down and absorb the words they are writing. I initially ran into the idea of copy work while reading Charlotte Mason's books on education, and have seen it put to use in a structured way in First Language Lessons for the Well Trained Mind, a book I use heavily for Language Arts. There is really no need to follow a set regime though - right now we are on summer break, which for us is like "homeschool lite" - math facts, reading, read alouds, and a bit of writing and copywork. I just find a good poem and write it out carefully in cursive for my 7 year old (who is just starting writing script) and have her copy it a few lines a day, or however she would like to do it at her own pace. Often by the time she has copied it, she has a good appreciation for the poem and sometimes has already started to memorize it a bit. Which leads me to...
You don't have to be Classical or Charlotte Mason inspired to benefit from poetry recitation. Forgive me if I go a little Anne of Green Gables on you, but there is a certain simple joy in being able to call up verse from memory. Young children will even spontaneously memorize their picture books without even trying. All it takes really is re-reading the poem several times over the course of a week and reviewing now and then. When a poem is mastered, we will usually make a big deal out of the child reciting for Dad, and also reciting for the video camera to delight the gradparents (or really more likely make them think that I'm really weird and trying to have little Anne of Green Gables children - but whatever).
Specific Literary Devices / Unit Studies
I taught an introduction to poetry class to kindergardners (yes, kindergartners) in our homeschool co-op a few years back. It was a little nuts, I admit, but I had a great time and the kids seemed to enjoy it and really get into many of the activities I had them do. In a circle-time set up, I read the poem out loud to the group with enthusiasm, then very briefly discussed one major literary device that the poem used. I touched on vocabulary words that would be especially obscure, and for some of the poems included a brief picture study to help illustrate the vocabulary. Then of course we did a hands on activity (usually some kind of craft) rounding everything out. For example, we made little styrofoam boats when we studied "There is No Frigate like a Book," and acted out a cozy house scene complete with having animal crackers and cocoa when reading "Animal Crackers." You could easily build a poetry unit study at home using the same model. Here are some of the poems I used, and what I emphasized:
"The Owl and the Pussycat" Edward Lear - nonsense words
"There Is No Frigate Like a Book" Emily Dickinson - simile
"maggie and milly and molly and may" e.e. cummings - descriptive language
"Macavity: The Mystery Cat" T.S. Eliot - narrative poem
"Animal Crackers" Christopher Moreley - rhyming
"Something Told the Wild Geese" Rachel Field - loud / soft
"The Children's Hour" Henry Wadsworth Longfellow - point of view
Poet Unit Study
Another idea is to focus on one particular poet. I have not done this in a structured way yet, although we have read several poems by the same author and commented a little (like, "oh, I love the way that Christina Rosetti writes about nature!" - but not much beyond that). At this point in my kids' education, I prefer to focus on the language of the poems themselves over the author, but I can see focusing on the poets as an interesting tie-in to history, or as an interest-led project. An older student might enjoy reading biographies of poets, and could practice expository writing discussing a particular poet's style and literary contribution. I think for most students, this can happily wait until middle to upper grades.
I hope this has given you some ideas to easily introduce your students to the delights of poetry! Happy reading!