Many new poets have become readers of Poetic Asides since when it began more than 10 months ago. And with close to 300 total posts, it's not a good idea for me to expect you to dig around looking for helpful publishing information. So, I'm going to give a real quick Poetry Publishing 101. (If you find it helpful, I suggest bookmarking this post.)
Before you attempt any publishing, you need to read a lot of poetry and write a lot of poetry. I put reading a lot poetry first--and by reading poetry I mean reading poetry by contemporary poets--because this is truly the best way to learn how to write effective poems. Successful poets pay attention to what they like in poems and spin it around in a new direction. Of course, you should also write--daily, or at the very least, weekly. If you frequently go longer than a week without writing, you might want to try setting up a writing routine or even reading more poetry (because reading poetry often sparks new poetry).
Avoid rushing into publishing before you've worked on your craft for a while. For instance, I worked on my poetry for more than 12 years and wrote thousands of poems before I felt comfortable enough to try getting published. Even after that lengthy apprenticeship, I've still had more than my share of rejection slips. The competition is fierce, so to spare your ego (of rejection) and your bank account (of postage expenses), I recommend you exercise a little bit of patience in your pursuit of becoming a world famous poet.
When you think you're ready to get published, start off by submitting to magazines and journals that accept poetry. Too many poets come to me asking how they can get their whole collection of poetry published when they haven't even published a single poem. (Of course, it should be noted that this is a natural way to think if you don't know the business of poetry publishing--so don't feel bad if I'm describing you.)
If you're not sure where to find magazines or journals that accept poetry, then I suggest checking out the most recent copy of Poet's Market. (Full Disclosure: I work on Writer's Market and recently have been going over pages of Poet's Market--and I edit the resurrected Poet's Market newsletter. So, yes, I'm a little biased to which reference I direct you.) You can pick up a copy at your local library or bookstore--or you can order online at http://www.fwbookstore.com/product/1538/23.
In this guide, you'll get more than 1,600 listings for magazines and journals, presses, contests, workshops, etc. But even more important for the poet new to publishing, it is loaded with practical articles and interviews that show you how to properly submit your poems.
If you've already been published in several journals and think you have enough poems to put together a collection, the best way to get that collection published nowadays is through poetry book and chapbook competitions. Chapbook competitions tend to be for collections of less than 48 pages (usually 24-40 pages is the norm), while full book length collections trend over this 48-page threshold. Neither type of competition is easier or harder to win--so don't enter the chapbook competitions thinking it'll be a cakewalk because the size of the manuscripts are smaller.
Of course, more and more poets are bypassing the traditional means of publication and doing it themselves. This tradition dates back as far as any poet can remember. Even America's great poet, Walt Whitman, was a self-publisher. But if you decide to go this route, make sure you can look yourself in the mirror and say that you're self-publishing for the right reasons. Don't do it just because it's the easy (or lazy) way of getting published if you actually want to build a readership over time. While saying you've got a book published can feel fulfilling, it loses its luster if the only people who own a copy of your poems are you, your mom, and your garage.
Finally, I'm not gonna get into the whole can of beans with those FREE poetry contests you can find in the backs of magazines and online. Not in this post. Instead, here's my account of my first publishing experience before I decided to get patient (that's right I was full of ambition at 16--and learned a valuable lesson as a result): http://www.writersdigest.com/poeticasides/Im+Coming+Out+Of+The+Closet.aspx.