Best Classical to "broaden" listening arena?

Difficult to know quite where to begin but here's one thought.
Get Pandora or some other such device and enter in piano concertos, flute sonatas or some such thing. Sonatas are usually one instrument or with an accompanying one. Concertos are usually one instrument with an orchestral background and just listen. See what catches your ear, is it the pacing or the playing and follow where that leads you.

Start with the "classics" such as Bach, Mozart, Beethoven. From there you can move on to Brahms, Tchaikovsky, Rachmaninov. These are the composers for cutting your teeth and the ones we all go back to for comfort. Then, when you are ready, treat yourself to some modern and contemporary music, but you need a foundation first to really appreciate newer music. Avoid Berg, Webern, Schoenberg. Very few have the patience or ears for their work.

I definitely agree with Uru975 on trying out a few things on Pandora first. It's a good way to narrow down a style. I used to play trombone, guitar, and piano and was exposed to different composers/styles through the other players in the bands/orchestras/small ensembles in which I played. Pandora could help you get a well-rounded exposure, and THEN you can hunt down the best recordings of things you like.

I'd start by deciding if you want solo piano, small groups, chamber orchestra, or full on symphonies. One of my favorite symphonies is Dimitri Shostakovich's 5th. I'd get Leonard Berstein with the New York Philharmonic. It ranges from light to heavy, so it should give you a broad introduction.