It took me a while to get used to classically trained "operatic" singing, but it was worth it. It didn't take me quite as long to get used to the opera being sung in a foreign language (probably a good thing, because I've never liked operas that are in English).
Once you get past those hurdles, and assuming that you have a modicum of interest in classical music in the first place, the next thing to do is find a composer whose work you like well enough to really, really get into it. All composers have a recognizable "style"...just as you cannot mistake the Beatles when they come on the radio, so you should be able to recognize your favorite composer in the same way
As for me, I chose to focus on the work of Richard Wagner. More specifically his 4 opera cycle, "The Ring of the Nibelung". It's a sprawling and intimidating piece of music and requires a lot of attention. But little by little you start to get a feel for it. The way the vocal melody lines play with the orchestra, they interact to a degree you won't find often in other operas. The way Wagner conjures up darkness and gravity through his music. The recurring themes (called motifs) that are designed to help tell the story through musical symbolism. As for those vocal melodies, they are everywhere and rarely repeat themselves, so you have more of a running dialogue rather than a collection of songs. As such, I don't recommend "without words" recordings for anyone who wants to get really serious with the music. The casual listener will find much to enjoy in the instrumental releases, but if you're gonna go for the gusto you have to skip to the real meat and potatoes, no matter how difficult it may seem or how sure you are that you won't like it. You might be surprised.