NPR has posted the cast recording to The Book of Mormon, the Musical.
While I recommend you listen to the whole thing, if you won't here's where the individual songs are found. To get the best gist of the play, listen to "Hello!" then "All-American Prophet" followed by "Making Things Up Again" next "Joseph Smith American Moses" and finally move to 3:16 of "Tomorrow is a Latter Day" to finish the story.
"Hasa Diga Eebowai," "Spooky Mormon Hell Dream," and "I Believe" are pretty spectacular, too, but the first is purely funny and the other two are about a character rather than the mission he's on.
I really don't want to travel to New York, but I'd really like to watch this play.