Sharon, there are actually dozens and dozens of fine literary magazines that publish poetry. Some of the more popular ones would include The New Yorker and The Paris Review, among others. Not only would places like this give you great exposure, but they are also going to pay you some money to publish your work.
But, not everyone gets into publications such as this. It takes talent, luck, and probably some strong connections would help as well. If you think your work is good, I'd say give it a shot, but don't get your hopes up too high.
If you go to a website called Duotrope's Digest, you can also use their poetry search engine to find other publications you can submit to. Go to the site, click the tab that says "poetry," fill in the search criteria, and then click "search." You'll get the names of the publications, the genres they publish, length requirements, pay scales, and whether they are a print or electronic publication. It's a great resource, and you can use it for free, although they appreciate donations.
Getting into a literary magazine is a bit of an art. (This applies to everyone, not just poets.) Different publications have different preferences. So, you can submit often to many different publications in the hopes that you'll accidentally stumble on a good fit. Or, you can take the time to research the pubs and only submit to the ones that seem to like your type of work. Seriously, either way works. I tend to take the latter approach, but I know a poet who takes the former and has quite a bit of success. You can decide if you want to spend your time research or submitting--both are kind of tedious, at least to me.
When you find a publication you are interested in, go to their submission guidelines. They will tell you how to submit your work to them. Often with poetry you can submit multiple pieces at the same time if you have them.
There are a few other things you should know about submitting, including the value of online versus print publications, multiple submissions versus simultaneous submissions, and the cover letter. Check out a few of our other posts, to get more information (1, 2, 3)
For me, the key to submitting is to expect failure. That may be a bit depressing, but getting an editor or agent to like your work depends on so many factors that are out of your control. If someone turns you down, don't necessarily blame yourself. Yes, sometimes, your work might not hold up. But, sometimes, the editor could have just gotten a parking ticket and simply isn't in the mood to make anyone happy that day.