True, but Paisiello wrote a version of the Barber in 1782, and the Beaumarchais plays that these works were based on were written in the order: “Barber” first, “Marriage” second.
I’ll second Puccini’s La Boheme. Paris, young penniless artists, love, tuberculosis - what more do you want? 
Tosca is good, too.
Wagner’s Ring can be a bit much to start with. Instead, try Tannhäuser (has a number of operatic hit tunes), Lohengrin (as previously mentioned), or possibly Parsifal.
I, too give high marks to La Boheme.
Two more great operas are Madame Butterfly (think Miss Saigon) and Der Fledermaus (The Bat). Fledermaus is sort of a much ado about nothing storyline that takes place during a masked ball. It’s a very humorous opera.
Good luck deciding! After you’ve attended, let us know what you thought.
Thanks for all the suggestions! I’m going to propose Aida, since it’s coming this spring. I’ll keep my eyes open for M Butterfly and Magic Flute.
I’m not sure I’m ready for Wagner.
It was Mark Twain who pointed out that Wagner’s music is not as bad as it sounds. Give it a try.
Another vote for Brother Mozart.
Oh, I love Mozart, and have the Magic Flute on CD. If that comes to town, I’m there. Pa-pa-pa-pa-gena.
The Ingmar Bergman film of “The Magic Flute” is just delightful. Unfortunately, it’s not the complete opera, but it is great anyway.