I made this list of basic gotta watch operas for another thread. It is by no means a comprehensive guide to opera, but if you've seen all of them you have a pretty good introduction to opera imo:
Baroque opera:
L'Orfeo -Monteverdi (Italian)
Rinaldo - Handel (English)
Classical Opera:
Alceste - Gluck (Italian or French)
The Magic Flute - Mozart (German)
Romantic (pre-1850ish) opera:
Fidelio - Beethoven (Italian)
Lucia di Lammermoor - Donizetti (Italian)
Faust - Gounod
Opera (Post-1850ish)
A Puccini Opera (pick a plot that sounds good, my rec: La fanciulla del West)
A Verdi Opera (pick a plot that sounds good, my rec: Macbeth)
A Wagner opera (My rec: The entire Ring Cycle, or just Siegfried)
Salome - Strauss (German)
Modern (early)
Lulu - Berg (German)
Pelleas et Melisande - Debussy (French)
Bluebeard's Castle (Hungarian)
Oedipus Rex - Stravinsky (Latin)
Modern (later) (this is just bonus points)
John Adams - Nixon in China (English)
Phillip Glass - Einstein on the Beach (math)
Peter Maxwell Davies - 9 Songs for a mad king (English)
Close calls that didn't make the list in roughly chronological order:
Dido and Aeneas (Purcell), Orfeo ed Euridice (Gluck), The Marriage of Figaro (Mozart), Don Giovanni (Mozart), William Tell (Rossini), The Barber of Seville (Rossini), Pagliacci (Leoncavallo), The Elixir of Love (Donizetti), Carmen (Bizet), The Pearl Fishers (Bizet), pretty much any other opera by Verdi, Pucinni or Wagner, Ein Deutsches Requiem (Brahms) and Verdi's Requiem (which are technically not operas, but still considered operatic works), Der Fledermaus (J Strauss II), Elektra (R Strauss), BORIS GODUNOV (Mussorgsky, highly recommended), The Snow Maiden (Rimsky-Korsakov), The Nose (Shostakovich), The Rake's Progress (Stravinsky), Moses und Aaron (Schoenberg), Wozzeck (Berg) Le Grand Macabre (Ligeti), Doctor Atomic (Adams)
These are just a few essentials I can name off the top of my head. The operatic repertoire is YUUUUGE, so to speak.