Do you like instrumental music at all? If not, that's going to be an uphill battle, IMO. Some metal fans like the "drama" and dynamic range of orchestral music. Others who mainly listen to guitar will pick up recordings of small ensembles or listen to "relaxing piano music" because they like mellow background music.
As for me, I didn't appreciate classical music until I had to play it in school, and my tastes have changed wildly over the years.
If you do enjoy instrumental music to some degree and want to expand your musical diet so to speak, I would recommend starting with popular pieces from respectable but lesser-known composers from the romantic and impressionist eras with a few more contemporary guys thrown in. [Warning: go too far into modern and you get experimental atonal noise I think they used to torture prisons at Abu Ghraib]
A big hurdle to me was burning through the songs that everyone has heard played one time or another in a commercial or a grocery store or a cheap music box/snow globe thing from Hallmark. That generally means no Mozart, no Bach, no Beethoven, no Pachelbel's Canon in D. All that stuff sounded (and still sounds) stale as hell to my ears because I have heard it a million and one times. Even Tchaikovsy, awesome as he is, usually doesn't do it for me because everyone and their grandma has picked through all his best stuff.
In my opinion, the most accessible are movie-score-like orchestrations with lots of drama and strong melodies. You could even start with Gustav Holst's Planets which inspired John Williams' score for Star Wars:
You've probably heard this one, but I don't think you can really ever go wrong with Wagner:
Edward Elgar and Malcolm Arnold are solid composers with a lot to choose from:
Here are some other names to explore:
- Franz Liszt
- Carl Orff - specifically Carmina Burana
- Igor Stravinsky
- Dmitri Shostakovich
- Aaron Copland
I also strongly recommend Maurice Ravel. He is my favorite composer and has strong works for both the orchestra and piano. He can write loud and aggressive, but his best work tends to be on the softer side of the emotional spectrum, IMO. Aaron Copland falls into that category as well.
Some more minimalist/modern composers who don't write straight noise:
Piano music that isn't merely elevator/hold/study music:
- FrƩdƩric Chopin
- Claude Debussy
At the end of the day though, you like what you like. I have tried time and again to rediscover jazz and have only found a few songs that I like. I just don't get the hype.