There are several ways to go about this.
1. The next time you are listening to a classical station, note what you've enjoyed and wait for the piece/performer to be identified. You can also check the station's website to see if there's an online playlist.
2. Check the schedules for your local PBS station and make note of upcoming classical performances. Tape, DVR, and/or watch them live.
The sad thing is that there appears to be less and less classical music, dance, and serious drama on TV, despite the proliferation of cable channels. All those things are still there, but you have to look for them.
3. Check out Classic Arts Showcase. I watch it on one of our county colleg stations in Maryland. It's clips of opera, orchestral, vocal, and instrumental soloist music, plus dance, jazz, and movie clips, all of which are worth checking out and something to flesh out your cultural experiences.
Classic Arts Showcase runs everything from extremely vintage clips (e.g., rare John McCormack) to recent (Bryn Terfel in his various recitals and operatic performances). It's an addiction.
http://www.classicartsshowcase.org/4. Go to the library and check out their classical CD/DVD/video collection. There's bound to be something in the standard repertoire -- operas by Puccini, chamber music by Mozart and Mendelssohn, etc.
You can obviously play around with music clips on all the various websites, such as the AllMediaGuide, Amazon, AllMusic, etc. There's also information on various composers and performers on those sites.
http://www.allmusic.comhttp://www.allmediaguide.comhttp://www.amazon.com5. Locate reference/introductory books (library, bookstore, etc.) on classical music, either for background information, suggested recordings, or enrichment. Some music stores will have a copy of the Penguin guide to classical music for reference purposes. It contains reviews of CDs and is a pretty solid guide, though obviously it's not flawless.
Ticket to the Opera by Phil G. Goulding is an enjoyable book for people who want to get their feet wet in the world of opera. It's got history, gossip, and a lot more.
6. Yes, go to local performances.
Good luck!