I've never heard Die Frau Ohne Schatten, by the way. I probably would have put Salome on a list for beginners
It's tough to choose just one opera by any of these guys. Salome may be a better choice for Strauss--it is, if nothing else, shorter. It's also more "modern," though.
Anyway, the woman without a shadow (of the title) is the Empress, who lacks a shadow because she is the daughter of the king of the spirits, and is immortal, and immortal beings in the world of this opera cast no shadow. The setup is that she has to come up with a shadow in three days, or her husband the Emperor will be turned to stone. Not having managed to come up with one by natural means (apparently, getting pregnant will net you a shadow, or so the implication goes), she resolves to go down among the common men and acquire one by trickery.
The opera contains MANY moments of surpassing beauty, of the kind that only Strauss could come up with (and, as with Salome, they stand out all the more being contrasted with what I'll call the "dramatic ugliness" with which a lot of the music for the unpleasant characters is written.) Strauss is in many ways a lot like Wagner in his sensibilities and attitude towards constructing an opera, and Die Frau is Strauss at his most Wagnerian in that sense. But Strauss was always a better composer, and it really shines through in this piece.
Also, Die Frau is one of the Strauss/Hoffmannsthal collaborations, and old uncle Hugo put his best work into this one, too. It's probably the one of their six operas (the other five being Elektra, Der Rosenkavalier, Ariadne auf Naxos, Arabella, and the inferior Die Aegyptische Helena) in which the two collaborators' input is equally prominent.
It's not performed all that often, because it's huge (and therefore expensive), and it's hard to stage convincingly. Many people find the story off-puttingly opaque, but those people aren't trying very hard, in my opinion. I also think the objection is out of date. In our modern age when fantasy/sf storylines are everywhere, why not in opera too?
I love Ariadne auf Naxos, too--long stretches of GORGEOUS music in that one. That'd be another good choice for a beginner (and it's funny, too).
(if I could have been arsed to come up with a specific performance.)
The one I have has Cheryl Studer singing the title role, and Bryn Terfel as Jochanaan. Terfel's great, as always.
I saw Deborah Voigt sing her first live Salome here in Chicago. She had lost, like, 75 pounds to play the role, and she looked great naked. Speaking, um, objectively.
--M.