Spend some time at your local library and on the internet, looking for contact information for literary magazines. Visit their websites and read their submission guidelines, taking note of the types of poetry that they accept as well as whether they pay for poetry that they publish or whether they simply offer free contributer's copies. Send in your work to those which seem most promising, making sure to follow their guidelines and that you address the appropriate editor.
Send submissions to a number of different magazines, making sure that each submission contains different works. If the magazine's guidelines specifically state that they accept simultaneous submissions then you can send them the same work that you have sent elsewhere, but make sure that both parties know about it when you do.
If you're wanting to try and get a book of poetry published, start by compiling a 20-25 page chapbook of your best work. Many publishers are much more willing to print a chapbook from an unknown poet than a full-length poetry collection, simply because chapbooks tend to sell much better than larger collections unless the collection comes from someone who is famous for their poetry or otherwise well known.
Research publishing companies in much the same way as you did magazines, focusing on small publishing houses and university presses to start with. Make sure that you follow the submission guidelines exactly.
Continue shopping your chapbook around to different publishing houses while continuing to write more poetry and submit individual poems to magazines. This will help to ensure that you keep developing your craft and that you continue to enjoy writing your poetry even while trying to get it published. The work that you put in now will pay off when you start receiving acceptance letters from magazines and start to get chapbooks and possibly larger collections published.