See if your library has Leonard Bernstein's Young People's Concerts series available for check out. These concerts were originally directed towards older children and early adolescents, and while, at times, Bernstein can seem a bit condescending by today's standards, they remain, by far, one of the best introductions to the orchestra, to musical forms, to basic theory of Western music, and to orchestral literature I have ever heard.

Wagner is opera, and not just any opera, but big, 19th Century, full spectacle grand opera. Mozart wrote operas, too, but smaller and lighter ones, both in their musical conception, and in their voicing for the instruments of his day. If you're listening to Bach operas, they're by J.C. Bach, not his much better known father, J.S. Bach.

As for particular recommendations, with nothing more to go on than you like piano and violin, you might help us a bit more by giving us some idea of period, or form. Good music for piano was written by many composers, including Beethoven, Brahms, Chopin, Liszt and hundreds of other major and minor composers. Concertos are one common form you might explore, and sonatas another.

The history and amount of music for the violin is mind boggling. Not only is there a mainline Western classical music tradition for the violin spanning more than 400 years, but there is an equally long and varied repertoire for the instrument, outside classical development, which nevertheless influenced classical artists continuously. Taking one side of the story, without the other, is know a lot less about either. But, start by listening to the great violinists of the 20th century play the standard classical repertoire. Look for recordings by Itzhak Perlman, Jascha Heifetz, Isaac Stern, and Joshua Bell.

If you want to do some reading...
A guide to classical music from the beeb.
A more concise guide.
A basic history of Western music development.
A glossary of terms.

Oh and to listen to clasical music shows on the internet you can start with BBC3 and CBC2.

for piano, i can recommend one of my absolute favorite pieces - erik satie's trois gymnopédies.

also, it might be worth checking out deutsche grammaphon's new webshop - they just started selling some of their ridiculously large back catalog as drm-free 320kbps mp3 files. judging from my sole purchase so far (of a cd containing a 1960s recording of holst's the planets that, for some strange reason, was paired with a recording of ligeti's haunting choral piece lux aeterna), the sound quality is pristine.

The NPR Guide to Building a Classical CD Collection : The 350 Essential Works

Listen here: Good classical station with lots of good info about the music