I'm another from the terrible "ancient music" crowd.
The first thing to learn when getting into real music--that is, classical or baroque, and even some "Romantic"--is that 'song' is usually the wrong word. The song as it is a rather recent invention. Classical music falls into all sorts of divisions and groupings. You might call, for instance, an operatic aria a "song", but that's not quite right. If you want to really grow in appreciation, you should at least have a rough understanding of musical forms. A quick Google reveals
this dictionary of musical forms. More simple Googling will find all sorts of information about music history, which will also lead to some composers worth hearing! Check Amazon for hints on what CDs to get of a particular piece--its classical music reviews are quite good.
Now, for personal recommendation, I must submit Bach. All of Bach. I am a tremendous Bach fan. I just got a piano rendition of Bach's Toccatas, Amazon ASIN "B000069CVA", which is simply awesome. However, this might not be quite your "thing"? If you're familiar with Beethoven, you might prefer somewhat later music. You could fill up your Beethoven collection with all his symphonies and piano concerti.
Another personal recommendation is for sacred music. Most of the great composers wrote at least something for the church: in chronological order, I recommend lots of Vivaldi; try the "Sacred Music" CD series put out by Hyperion records, which I love deeply; Bach's Mass in B minor, among his many other sacred works; Mozart's famous Requiem and his other Mass settings; Verdi's Requiem; oh, the list goes on: Berlioz, Rutter... My tiny library is 150 hours long, and I feel that I have nothing. Listen! If you like it, get more of the same form, or by the same composer, or by similar composers.
"The best of" or other compilation CDs are nice to get a taste for a composer and his various forms, but once you
have gotten a taste--by all means go ahead and get full performances of the music you like. Handel's Messiah, for instance, has a few grossly overplayed sections ("Halleluja"), which have almost completely lost their beauty, along with a lot of really delightful pieces, say, in the third section, that are rarely played.
It's fun! Don't be too serious about it. Listen, listen, listen. Get headphones or good speakers. You can get the cheap Sennheiser models for a pittance; classical music really demands high quality. Don't get turned off by the stigma of stuffiness or "old"ness--it's not! I say this as a teenager.
Oh. And go to concerts too. It's not that expensive, really, especially if you qualify for student discounts, which are usually _extremely_ deep. This depends on where you are--here in Los Angeles, we have a good Opera and Philharmonic, and lots of other musical opportunities--for instance, weekly organ concerts at the Cathedral after Mass.
Fun, fun, fun, fun, fun! Then try playing it! That's for another post though.