I would suggest the following:
Symphony No 3: Bernstein on Sony. He talks about the piece for about 10-15 minutes. This was the piece where Beethoven starts to really break from the past and the commentary is interesting if that attracts you. Excellent version too!
Symphony no 5/7: Kleiber on DG. This, in my opinion, is one of the best discs ever made in the history of recording. No5 has the famous begnning and both have glorious melodies.
Symphony no 6: Beautiful music. Walter conducting is a popular choice.
Symphony No 9: Lots of choices here. I like Karajan (the 63 version), but there are many good versions.
Piano sonatas: Lots of good stuff. You could choose something famous to start like the Moonlight. Again, many good versions including Schiff, Brendal, Kempff, and Rubinstein
Piano Concerti: There are 5, but the last two are particularly good, I like the version with Haitink conducting Parahia.
4 Symphonies. Lots of good versions, including from Karajan (not the last one in digital, the one in ADD), Abbado and Mackarras.
Carmen and L'Arlesienne Suites: Tuneful. Dutoit is excellent.
There is a one disc of Chopin played by Rubinstein. It is excellent. Of course, you can buy all the piano music for about $20-25 as well.
Sym No 9: Kubelik or Mackarras are good (as are many others).
Peer Gynt Suites. SUch melodies! I like version with Jarvi on DG (but don't mix up with complete piece). This version includes other suites that are also great!
Water Music and Royal Fireworks: Always liked the Kubelik version, but many others too.
So much here. I'll let others suggest their favorites. You might want a type of 'Greatest Hits' to see what you might like.
Rimsky-Korsakov: Scheherezade. Wonderfully beautiful at times and powerful as well. Mackarras on Telarc is excellent.
Rossini: Overtures. Many good versions. Would suggest Orpheus Chamber Orchestra. The overtures are a good introduction as you may recognize some music used in cartoons and they are short pieces.
Tchaikovsky: You may be familiar with the Nutcraker, and that may be a good place to start. Mackerras is good.
This barely begins to touch the surface. Some other pieces you may like:
Hadyn: Later symphonies.
BAch: Keyboard concertos.
Rachmaninov/Tchaikovsky Piano conertos (usually #2 and #1 respectively).
Schumann/Grieg Piano Concerto (often paired together).
Elgar Cello Concerto (DuPre is heart rending).
British Light Music, Hyperion (volume 1, there are 4). This is just happy music, well played. Lots of pieces and different composers.
Dvorak Slavonic Dances (Kubelik is good here). Very catchy music. I always sing it after I hear it.
Brahms: Both Piano Concertos.
And for opera, I'll recommend two accessible works:
Puccini, La Boheme. Version with Pavoratti and Karajan conducting is ideal starting place. 4 acts, 30 minutes each. You may recognize some of it.
Humperdink: Hansel and Gretel. You know the story already, and music is great.