Which Shakespeare play should elementary children do first?

Oh, it's gotta be Hamlet. Father killed, mother involved, vengeful
ghosts, poisoning, murder, treachery and double crossing, insanity,
suicide... how can it miss? It's about as gory as any fairy tale, what
with handy woodsmen slicing open wolves and witches. Macbeth would be
a close second.

For real blood and guts, Titus Andronicus can't be beat. A horrid play
(his first) and truly a bloodbath. The little tykes wouldn't sleep for
a week.

But seriously, Twelfth Night is funny and mild, yet more for a
slightly older set. Comedy of Errors is the shortest of Shakespeare's
plays (a nice consideration for kiddies), but not really suited for
that age group.

I believe Midsummer Night's Dream is the best for them and likely
would need little or no editing.

If by "do" you mean "read"...

Have a look at the "Shakespeare Can Be Fun" series:

Information:
http://www.fireflybooks.com/Kids/Shakespeare.html

Selection at Amazon:
http://www.amazon.com/exec/obidos/tg/stores/series/-/90497/ref=pd_sr_ec_ser_b/102-4325756-8802560

They seem to be nicely adapted, switching from the original complex
meters to a friendly-to-the-ears rhyme.  They're also converted from
play format into verse, which is good for kids who just want to read
the story, but bad for kids who want to actually perform.  Also, the
books appear to be geared toward upper-level elementary students (I
doubt they'd be useful to anyone below 4th or 5th grades).

If by "do" you mean "perform", then you'd have to try to find some
kind of adapted/simplified work.  Shakespeare in original form is just
too complex for even the brightest elementary school students to fully
grasp; sure, they could be made to memorize the lines, but there's
little fun in performing a play that you don't even understand.