The easiest way to start is to get a book directly from the author (like the one attached -- it is the final proof, so still has some typos, undone fact checking, etc.) or from the library (I'm assuming Yale buys tons of new poetry books). *Query* places which publish reviews, starting with places the author's poems have appeared (of course they will want a positive review), mentioning that you have a review copy. Make it an open query -- I have this, I am a beginning reviewer, I am open for assignment, I would like to receive your review specs (length, style book, etc.)
Write the review "on spec," without waiting for a response. Hopefully, by the time you've finished, someone will respond. If not, start circulating it for publication. Online has a faster turnaround.
After you have placed the review, ask the REVIEW EDITOR if s/he will assign you another book to review. You will probably have to ask. Do not be daunted if there were tons of editorial changes (for example, Boston Review capsules are evolving to become more theoretical and less journalistic as Tim Donnelley (the poetry reviews editor) proceeds to his PhD; Chicago Review likes "freshman comp five paragraph themes" rather than review formats, etc.). If they've made tons of changes, know they probably do that to everyone; they might be glad you know what to expect after having been through the process.
Whether or not this happens, send e-mails and letters to other (better) review organs, using the review you've just had accepted for publication as a sample. Don't wait until it actually appears in print (the wait can be a year) -- just paste it / print it out. Editors don't expect xeroxes of the actual printed page or links.
AND send e-mails or hard copy to the PRESS: i.e., this is the review I just wrote of one of your books, this is where it was published, I would be interested in reviewing more of your books. This is the crucial step. You want presses to just start sending you their books. I.e., Copper Canyon sends me everything they publish. You can always place them SOMEWHERE.