If I find a poem online through Google, is it considered in the “public domain?”
No, something being on Google definitely does not put it in the public domain. The Library has a Web site that contains links to information about copyright and fair use at http://library.uhd.edu/copyright. It includes a link to a public domain chart at Cornell, which notes that works published 1923 and after are not in the public domain if they had a copyright notice.
Next, you would want to consider if you could claim "fair use" on your use of the poem. The page I linked above includes a look at the four factors that determine fair use. There is no difinitive answer to your question, but you would want to consider: is this for non-commercial, educational use? Also, the amount of the work you use is important. There are no absolute guidelines, but if you wanted to include the whole poem, you would need to consider this: “The more you use of a given work, the more you should be prepared to argue that the amount used was critical to your educational objective.” You’d need to consider all four fair use factors, though.
If you want to request permission to reprint the poem, try to look it up at the Copyright Clearance Center http://www.copyright.com. This organization gives you some avenues to request permission to reprint works under copyright.