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.
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.