It really depends on what you want to get out of poetry, in all honesty. Functionally, poetry has done a lot of different things from culture to culture.

Given that you're a story man, you may want to give the epic poets a try, assuming that you can find a good poetic translation of Homer, Virgil, or Dante. Even much of Chaucer is poetry, though the Middle English can be tricky. Milton's Paradise Lost is definitely worth the effort and can be read by anyone who can read Shakespeare.

I'm a big fan of G K Chesterton's poetry, though I know it only piecemeal. He has poems from the humorous (eg. "The Song of the Strange Ascetic"), to the historical (eg, "Lepanto") to the religious (eg, "The Donkey").

Probably the most influential poet on my own work is Edgar Lee Masters. His Spoon River Anthology tells what's really going on in a small American town by having the citizens speak from beyond the grave. Several events are narrated by various of the deceased, and you as the reader can only piece what's going on by seeing several perspectives.

And of course there's haiku, but that probably deserves its own response. If you can find a copy of Blyth's 4-volume collection, that's where I really started appreciating the genre. William J. Higginson's Haiku Handbook has the advantage of being only one volume.