Well, there's so much good stuff that it's hard to know where to start. On the plus side, the nice thing about classical music is that a lot of the obvious stuff is also good.
Beethoven, Bach, and Mozart spring to mind for the older stuff, Ravel, Debussy, Mahler, Nielsen, Bartok, Shostakovich, Britten, Stravinsky and Webern for the late 19th - mid 20th century.
A random few personal picks would be the Bach B minor mass and unacompanied cello suites, Mozart 19th piano concerto 19 (iirc) , Beethoven 7th and 9th Symphonies or 4th piano concerto, Brahms 4th symphony, Mahler - Das Lied Von Der Erde, Ravel - Mother Goose Suite, Stavinsky - Petrushka, Bartok - Rhaposodies for Violin and Orchestra, Messaien - Quartet for the End of Time. But that's a pretty wide range of styles, so getting some idea of what appeals to you is probably a good idea.
As for how to listen to it - I find that to really appreciate something properly you have to either see it live or sit down somewhere non-noisy, turn off the lights, and listen to it loudly (classical CDs tend to have none of the compression of pop, so the quiet bits really are quiet). On the other hand, listening to it as background music seems to help me to get familiar with something and then get more out of it when I do listen properly.
To get an idea of roughly what you like on the cheap, you can find a lot of good stuff in charity shops for one or two pounds a throw, or (if you're in the UK) just stick on Radio 3 and see what they're doing... on the other hand, once you've got something you like, it's probably worth doing some research to find a really good recording, either by checking reviews or one of the big guides (eg the Penguin Guide to Compact discs, which a lot of classical shops will have hanging around anyway) or by just asking the people in the shop.
Oh, and if you're anything like me, you might find that there's some stuff that you only 'get' once you've 'got' stuff that's a bit similar to it, particularly in old classical and baroque and very modern music, which often take a while to come to terms with the limitations or lack of limitations of the style, whereas romantic / post romantic / impressionist stuff is often fairly straightforward for a modern listener. So don't write stuff off if it doesn't grab you immediately.
Christ, that was longer than I anticipated. Sorry if any of it was overbearing / patronising...