Small press magazines are an ideal starting point for poets to start submitting their work.
Many famous poets saw their poetry published for the first time in small press magazines. For an unpublished poet nowadays it is unrealistic to expect to have a whole body of work published by a major publisher.
Today, sadly, that'd be true of the major poets of the past as well - as major publishers are unlikely to take risks. A well-known celebrity who writes dreadful poetry will more likely be published, because a big fanbase will probably guarantee sales.
But, don't despair. Being published in small press magazines will slowly but surely build your reputation, and you will be able to hone your craft. The better your work, then the better and more renowned will be the magazines accepting your work.
A good guide for a list of small press poetry magazines is 'Poet's Market', which is reasonably priced, though it may be available in your local library. In 'Poet's Market' you'll get more of an idea of what publications seem the right places to send your poetry to. Properly researching magazines to submit to will save a lot of wasted time in the long run. Identify your specific interests and strengths regarding your poetry, i.e. style of verse, subject matter, etc.
ALWAYS remember, when submitting your work, to send a stamped, self-addressed envelope, and when submitting to publications outside your own country, remember to enclose enough IRCs (International Reply Coupons - available at most post offices) to cover return postage.
Never send complete manuscripts, because, remember, these are small press magazines. It is best to send no more than half a dozen poems, and check if a publication only takes original material, before sending the same poems elsewhere. Always, obviously, keep copies of your work.
Editors will obviously be subjective, so you may sometimes find work that you consider among your best, rejected, and poetry you consider to be inferior, accepted. But always try and send in work of a high quality, and not some of your weaker work to very small publications. If you gain an early reputation for quality work, then you'll have a better chance of being published in the same publication more than once, and, conversely, send in average material, and an editor will not be very positive towards your work in future.
Finally, check the guidelines of each publication thoroughly, and make sure your work is neatly presented, with good grammar and spelling.