There is a difference between self-publishing (where you pay everything yourself) and shared expenses where you are accepted by a publisher but pay part of the expenses. In the latter case, they will help you with promotion. My husband did this and his book is available on Amazon and numerous magazine articles have been written in the local and specialized press. I don't know if his publisher does poetry but if you want their name, let me know. They are reputable and have been helpful.
You can, as others have suggested, submit your work to publishers and/or literary agents. You must be prepared for a great deal of rejection, first of all. Secondly, you must understand that book sales are down, magazines are very thin (decreased readership + decreased advertising = fewer pages), so they reject the majority of what is submitted. Many magazines refuse to even read unsolicited submissions. Also, literary agents charge a fee and there is no guarantee of publishing - it's very difficult to evaluate their qualifications and their influence. There are some wonderful ones, to be sure, but they have a limited number of hours and works they can represent.
You might look at the Independent Publishers - that is an organization of many publishers, including regular independents and university presses. Univ presses often lean toward their own alumni or professors - if you have a connection that way, you might use it.
Also look into poetry readings in your area - many have open mic nights and they tend to attract newcomers as well as people with experience. You might network through them. You could also call children's libraries and offer to read during story hour - you might attract some attention.
Try to evaluate what you really want - do you want to see your works in print just for the satisfaction of having something to hold in your hand or to give to friends, or do you want to attract a broader readership and earn a living? These are different philosophies and require different approaches.