As a pianist and music teacher, I'm glad to see someone interested in learning more about classical music! I would recommend finding a compilation CD that has music from the four general periods in music history--Baroque (1600-1750), Classical (1750-1825), Romantic (1825 - 1900) & 20th Century (1900 - present). Once you do this, you can see which period appeals to you the most. At that point, find some CD collections of music from that time particular time period.
There is a wealth of incredible music out there to choose from; however, a new classical music listener can't go wrong listening to (Ludvig von) Beethoven. His early music is considered Classical period while his later compositions are generally categorized as Romantic. (Johann Sebastian) Bach's Preludes, Fugues and Inventions from the Baroque period are beautiful and interesting to listen to while Mozart's sonatas and opera music are probably the best choices from the Classical period. Chopin, Tschiakowski and Debussy from the Romantic period are also good choices for new listeners. When it comes to 20th Century composers, you might want to give Gershwin or Copeland a try if you prefer more conventional types of music. Ives and Stravinski are also wonderful, but a bit more progressive if you are the adventurous type. Phillip Glass is an excellent composer from the latter 20th century, as well.
As far as recommending specific CD's, I have a compilation CD that I love called "Piano Classics" published by St. Clair that contains a number of well-known pieces by Beethoven, Debussy, Liszt, Chopin, Mozart and Schubert (Classical & Romantic period composers). You might want to look for recordings performed by Ashkenazy, Alicia Delaroche and Vladimir Horowitz (piano), Isaac Perlman (violin) or any of the Symphony Orchestras from the major cities in Europe, especially the London Symphony. The Boston Philharmonic is also wonderful.
Best wishes! I hope you enjoy many hours of happy listening!