After Shakespeare and the Bible, the greatest source of quotations in Bartlett's is the great poets, including Milton, Tennyson, Byron, Keats, Yeats, and Eliot (as well as Emerson, the sole man of prose in this elite group). Of course, much of Shakespeare is poetry, as are parts of the Bible, notably the Songs of Solomon and Psalms.
In many cases, you will just want to use a few lines to put a memorable poetic light on your subject. To that end, you can just look up the name of the poet in the author index of The Quotionary and see what topics he commented on (Bartlett's has a subject index, but does not show the name of the source there, just part of the line in question, although it is more likely to have more of any given poem). Not all references for the author will be poems; some will be comments on the topic.
For example, in talking about the difference between self-esteem and arrogance, you might quote Milton's “Paradise Lost”:
Oft times nothing profits more Than self-esteem, grounded on just and right Well manag'd.
A couple of lines from Lord Byron's “Childe Harold's Pilgrimage” have been quoted to put nuclear war in perspective:
A thousand years scarce serve to form a state; An hour may lay it in dust.
If you have the time, you could quote poems more extensively for a more powerful impact (although you do not want to turn your speech into a poetry reading). It would help in these cases to put up the words for the audience to read along with you, since absorbing the meaning of more than a few poetic lines can be difficult, but avoid anything so cryptic that it needs a lot of explanation.
For example in discussing the need to have faith that times will change, Tennyson wrote, in “In Memoriam A.H.H.”:
O, yet we trust somehow good Will be the final goal of ill … Behold, we know not anything; I can but trust that good shall fall At last — far off — at last, to all, And every winter change to spring.
Yeats acknowledged the benefits of forgiving oneself in “A Dialogue of Self and Soul”:
I am content to follow to its source Every event in action or in thought; Measure the lot; forgive myself the lot! When such as I cast out remorse So great a sweetness flows into the breast We must laugh and we must sing, We are blest by everything, Everything we look upon is blest.