Brownsfan has a great point. The radio can be a great way to explore classical, especially if you have a public radio station w/o ads. If you live in the Bay Area, you're screwed in terms of classical radio- it's just awful. Most other major metropolitan areas, you'll find something.
Side note: Rachmaninoff is not a modern composer. He's towards the end of the Romantic, but still well entrenched in that tradition. Generally, unless you're talking about French Impressionism, the modern period is said to have started with Stravinsky's Rite of Spring. That might be a little much to start with- Rachmaninoff, Tchaikovsky, Brahms, Mozart, and Beethoven are are great choices. And how bout some Prokofiev? The Second Piano Concerto is just wonderful, as are most of the symphonies. The Fifth is the most famous, the Seventh is my favorite.
Side note: Rachmaninoff is not a modern composer. He's towards the end of the Romantic, but still well entrenched in that tradition. Generally, unless you're talking about French Impressionism, the modern period is said to have started with Stravinsky's Rite of Spring. That might be a little much to start with- Rachmaninoff, Tchaikovsky, Brahms, Mozart, and Beethoven are are great choices. And how bout some Prokofiev? The Second Piano Concerto is just wonderful, as are most of the symphonies. The Fifth is the most famous, the Seventh is my favorite.