I think that one of the best ways to discover the world of classical music is to listen to the radio -- either fm or online. Depending on where you live, you'll probably do better with online listening. Check out the web sites for any of these: KUSC, KUAT, WQXR, WHRB, WFMT. There are more, but that short list should get you started. You'll hear a wealth of good music in all of the different sub-genres. Discover what you like. Then research recordings.
That's how I got started many years ago, listening to WQXR-FM right at the edge of the reach of their signal into southern Connecticut.
-- John