I’m a fan of the big three: Pirates, Pinafore, and Mikado. In fact I have the D’Oyly Carte recordings of all of them, and have listened to each a dozen times or more. I enjoy them as much for Gilbert’s wry humor as for Sullivan’s eclectic tunesmithing. So, where do I go next?
I vote for *Patience or Bunthorne’s Bride.*It’s a parody of the aesthetic movement and the Pre-Raphaelites. It has one of my favorite G&S numbers, “Prithee Pretty Maiden.”
I’d suggest either Iolanthe or Ruddigore. I like the Sorceror very much as well. Be warned that the Yeoman of the Guard is the one G&S operetta that doesn’t have a happy ending.
Well, not exactly a consensus there. Five suggestions from three people. 
I think I’m going to try Iolanthe, but I’m not sure why. Because it’s first alphabetically? I’ll let you know how it goes.
I need to pick up a fifth opera, myself. I love Pinafore, Pirates, Mikado and Trial, but I’ve heard hardly any of the rest.
Trial by Jury is a short one, and worth it just for the judge’s song, where he describes the path of his career advancement (“I fell in love with a rich attorney’s elderly, ugly daughter”).
Not just that one - “Trial” is basically made of win. From the jury instructions, to the Mutton Explanation for Infidelity, to the collective infatuation with the plaintiff - you owe it to yourself to listen to Trial by Jury. And then see it live.
Ruddigore, followed by Trial By Jury.
I’ll second Patience.
Iolanthe has the Lord Chancellor’s Nightmare song, which makes it worth it.
Iolanthe is the true embodiment of everything that’s excellent!
I’d go with either *Patience *or The Gondoliers.
Princess Ida is obscure but has the most beautiful music, IMO.