I am not an opera fan...but...

Disclaimer---I am not much of an opera fan. However,I get the occasional jones for an opera fix or at least the feeling that I should be trying to like it. (Especially when I'm in my "godfather" mode) I recently bought a CD entitled "Simply the Best Night at the Opera. This one satisfies my needs. It is great.
Probably the easiest opera for an opera newbie is Puccini's LaBoheme. This is easily one of the most beautiful, easily assessible, melodic, sad, etc etc opera's, ever. Great duet's. Just turn up the volume and settle in. :-)
I'd recommend La Traviata by Verdi as a good starter.
is worth getting to know. Claudio Abbado's version with Vienna Philharmonic on DG is the one with which I'm most familiar. It was inspired by the spirit of the American and French revolutions. Lorenzo da Ponte, a defrocked priest, wrote the libretto while a fugitive just one step ahead of Venetian and Austro-Hungarian authorities, and Mozart wrote the score. Eventually, da Ponte escaped to the land of the brave and home of the free, and I like to think that Mozart would have followed him there, had he lived. Recommend you familiarize yourself with the libretto which is a critique of aristocracy.
NOTE: Above message is about Mozart's Marriage of Figaro.
One of the finest recordings of opera music I have ever heard is a relatively recent release titled "Arias for Farinelli", featuring arias that were performed by Carlo Farinelli (1705-1782), a castrato virtuoso. The arias are sung by Vivica Genaux, who has an extraordinarty coloratura mezzo-soprano voice, with accompaniment by the Akademie fur Alte Musik (of Berlin). The recording is on the Harmonia Mundi label, and is available in both CD format (HMC-901778) and SACD format (HMC-801778).

A number of my friends who are not opera buffs have raved about the recording (several even asked me to make CD copies), and after many listenings I still find it thrilling.
Newbee, I have ordered LaBoheme on your recommendation. Thanks for the suggestions guys. Maybe I should have named this thread "Opera for People Who Don't Love Opera " or "Opera for People who Know There Is Something to Opera but Haven't Figured It Out Yet"
I found with my daughter that the theatrical aspect of opera is what hooked her. She liked watching Wagner's Ring cycle with the sub-titles. And the Magic Flute. Then the music could stand on its own.
Papertrail, La Boheme is the very first opera which captured my heart, too. I can't call myself an opera buff, but I am very glad that Puccini passed this way. Each and every one of his operas offers lush, romantic, memorable melodies. Nessun Dorma from Turandot is probably his best known aria for tenors and you have likely heard it. However, now that I own all of Puccini's operas I will personally urge you to make Madama Butterfly your next acquisition if you are going to try any others. Pagliacci by Leoncavallo is another great one for neophytes. Being short it is usually paired with Cavalleria Rusticana by Mascagni (beautiful, too). All of the aforementioned are of the "Verismo" school of Italian opera which flourished from the 1880s until Puccini's death in 1924.

I would also suggest that you look into some collections of Puccini's arias as they offer the most concentrated display of his gift for melody. I can recommend a few if you wish.

Newbee was right to hook you up with Boheme and I do hope you come to love it as we do.

Personally, I think a great way to start is with Mozart, especially The Magic Flute. Der Fledermaus (Johann Strauss who in most ways I despise) is also really fun and very accessible.
Of course 'Die Fledermaus' (The Bat) is not an opera but an operetta ie. they talk as well as sing!
Aahh that brings back memories of working with a polish ensemble showing
'Die Fledermaus' on New Year's Eve (its a tradition) in Berlin! Don't think I was ever that drunk again as after that show! All champagne, vodka and polish folksongs until dawn.