Frankly, I think magazines and anthologies are a bad place to start. Not because the poetry in them is bad, but because the best way to get engrossed in, understand, and come to love poetry is by reading a book. A dog in a poem by Eliot is going to be very different than a dog in a poem by Michael Palmer. All poets have particular words, images, and ideas that they play with over and over. But you won't necessarily understand the particularities of an author if you read a single poem (most magazine formats) or a smattering that spans a lengthy career (anthology format). Poets organize their work in books. I think you get more out of reading a book than you do out of these more scatter-brained formats.
I'd start with "Crush" by Richard Siken. This book is exciting both in the musicality of the lines, and the content. (I know I gave "Praise Song" a good review, but it wasn't a very musical poem). And don't kid yourself, musicality is very, very important. In the days of prose poems and free form poems, the way the line sounds is the only thing that distinguishes "poetry" from a Cuisinart Instruction Manual. When you start reading poetry think about the way the poem sounds--the words, the line breaks, the pauses in the white spaces and between stanzas.
It should sound good.
Siken's book is divided into 3 sections (I, II, and III) and concerns itself with heartbreak--a powerful, if popular theme. But Siken plays with narrative so the cause of the heartbreak keeps changing. The overall arc goes something like this--first section, the speaker's lover has left him. Second section, the lover died. Third section, one lover kills the other. But within each section there is some play, some give-and-take with the absolute narrative. The speaker has been shattered. He can't tell us the narrative of how he got here, or at least he can't tell us with any degree of certainty. All he knows is the raw feeling. Don't worry if you don't understand exactly what happened between the lovers. The speaker doesn't know either, and that gives the book the feel of a mystery novel--you're looking at the corpse of a relationship, picking up the clues, trying to piece together a before and an after.
And Siken plays around with ideas of control: he is an actor in a movie, "We're filming a movie called Planet of Love"; he's a writer, "I'm just a writer. I write things down"; then he's a director, "I have a megaphone and you play along." But ultimately everything spirals out of the speaker's control. As with a great deal of poetry, piecing together a chronology is impossible and unimportant. What you're left with are the pieces of life that don't fit into the superstructure narrative: We met, then we lived happily ever after. That's most of life.
Read the whole book to the end. Enjoy the poems. It's a book where things are constantly breaking or blowing up, and then you get to look down at the glittering pieces. Move forward through it. Read slowly, but don't think too much about what you are doing until you get to the end. Then and only then should you look back.
When I finish a book of poems my first question is always what did I like? Then I ask myself why I liked it. What sounded good? Why? I re-read the poems I liked, often I re-read the whole book. Then I ask myself if there is something I feel confused about. Is there something about the book I don't like? or don't understand?--or don't like because I don't understand? Poems don't rely on narrative and plot the same way novels do. You aren't going to spoil them by knowing what happens next. Poems are meant to be read and re-read in a fairly short-time span. By the time I get to the re-reading I am usually so engrossed in the book I don't worry about much else.
We make demands of poetry we don't make of other art forms. We demand that a poem make sense, that it have a "meaning" below its surface, and that this "meaning" be universally identifiable (often on a first, sloppy reading). We don't demand that paintings have "meaning" in this way. We like the way a painting looks, so we hang it in our kitchen. We are much more willing to just enjoy the brushwork.
And poetry aside, we know how to enjoy and appreciate art. Songs (as Ashley pointed out) don't always make literal sense--"Lucy in the Sky with Diamonds", anyone? And yet we are able to enjoy songs without worrying about whether or not we are "getting the meaning." Then, through our enjoyment, we find meaning. "Lucy in the Sky" means a great deal to a generation of people. We need to apply more of this type of thinking to poetry.
And this brings me to my most important advice to new poetry readers: DON'T READ ANYTHING YOU DON'T LIKE. Of course it is vital, as I have explained before, to give poems time, and not to rush to judge a lyric you haven't engaged with, but once you've done that--DO NOT READ ANYTHING YOU DON'T LIKE.
Aesthetic appreciation is the easiest way into a poem. I really believe that poetry is FANTASTIC. I don't think enjoyment is a difficult mandate. Enjoy the way the poem sounds, because if you don't ever enjoy yourself, you probably won't read anymore poetry. That would be a shame--for you, and for poetry.