The usual place to start is with magazines. There are many small, regional magazines, and quite a number of national magazines. Some are just for poetry, others do short stories too. The magazines often contain reviews of recently published poetry books too, and it's worth reading them.
To find magazines that might suit your work, the best way by far is to go to a Poetry Library. There is one in London on the Southbank, and another in Edinburgh, near the Grassmarket. They should have recent issues from a range of magazines. Find a few magazines that print poetry you like, or which you think might be like yours. You can get the details of how to submit, addresses and so on, from the samply copy, but you should also consider subscribing to one or two. Subscriptions keep the magazines alive.
Almost all will say something like, submit a maximum of six poems at a time. I would say make it less than that. Double check for spelling mistakes - have someone else read them before you send them purely to check for that kind of thing - someone you know has very high standards of grammar and spelling. Print them as simply as possible on plain white paper, lines not too close, print a comfortable size. There should be nothing on the page but your name (with maybe an email address) and the title and the poem. Have your name and details in the header or footer space in a font slightly smaller than you use for the poem itself. Use a very ordinary font, like Arial or Times New Roman. Your poem has to persuade the editor on the strength of the words alone. Everything else is distraction and will work to your disadvantage.
Submit them with astamped addressed envelope for return, and a very simple covering letter. You don't need to say much more than 'here are four poems for you to consider for publication'. Even that is redundant, your covering letter is there just to give your name and contact details in case they are interested in publishing your work. If you write a paragraph or so about yourself or your work, it is only likely to actually be read if they've already decided to publish, and want to know a bit about you. It will not ever influence the decision about whether or not to publish. Editors get piles of submissions daily. They rip the envelope open, turn straight to the actual poems, and then usually push the whole lot back into the return envelope.
Expect rejections. You will inevitably get them. Everyone does. After a while you might start noticing comments coming back with the rejections - that's a good sign, because normally editors haven't got time to spend long over every submission. Eventually, if you listen, and keep reading poetry, and thinking critically about what you're doing, and thinking particularly hard about comments, you will start to get the occasional success.
For online magazines, it's the same basic deal, just without the envelopes. Keep things as simple and uncluttered as possible.
The other thing you can do to help your chances of book publication is to enter competitions. The Poetry Society runs an annual competition which is one of the biggest, and actually has reasonable prize money, but the advantage is not winning the money, it's the fact that editors will look at the winning (and shortlisted) poems. They have a page with information about several competitions, as well as their own annual competition. Competitions often cost money to enter. I think the Poetry Society one costs £5 for the first poem and then £3 for each poem after that.
It's a simple fact that until you have built up some sort of a track record by publishing in magazines, winning or getting placed in major competitions, no publisher will publish a book of your poetry. If you can get one or two poems into a book with work from other writers, jump at it, because it will all help build a case for your own first collection.
Another thing worth considering is readings. This doesn't have to be slam events - all sorts of places will either put on poetry readings or include some poetry in an evening of other stuff. I know a folk music club, for instance, that usually has an open mic spot at some point in the evening (and occasionally books an established poet as the main event). There's something about the thought of reading a poem out loud in front of other people that focuses your mind, quite apart from any feedback you get from the audience when you actually read.
Good luck.