It sounds like you prefer non-sung through musicals. Of the options you've listed, I've seen School of Rock, Chicago, Phantom of the Opera, and Book of Mormon. With the exception of Phantom, I've only seen the Broadway productions (not the touring versions), so keep in mind that your experience may be somewhat different.
I found the currently touring, "re-imagined" version of Phantom to be very disappointing; I recommend that you hold out on that one until you can see "the brilliant original" on Broadway, as it's a far superior production. I believe every theatre fan should see this show (meaning the original) at least once. In any event, be aware that Phantom is almost completely sung through, if that's off-putting to you.
School of Rock is a bit of a departure for Andrew Lloyd Webber in that the score is not orchestral and the show is not sung-through (there's spoken dialogue between the songs, which have clear start and end points). This is a classic story of a ne'er-do-well who makes good in the end, so there's nothing particularly new or innovative about it, but it's a lighthearted, upbeat, good-time show with brilliant child musicians who all play and sing live.
Chicago is a musical theatre staple with some pointed observations about the society we live in. It's sexy and darkly humorous, with a biting wit. Like Phantom, every theatre fan should see this show at least once.
Book of Mormon is crude and irreverent and uproariously funny. However, those who are easily offended by swear words and sexual and religious humor should probably avoid it. I'm about as liberal and non-religious as they come, so I thought it was hilarious, but I was quite surprised that no one around me walked out of this show. Even in New York, I'm amazed that there are still enough hard-to-shock people who haven't seen this show to regularly pack the house.
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"Michael Riedel...The Perez Hilton of the New York Theatre scene"
- Craig Hepworth, What's On Stage
