Salmo,
I agree with Ryley that Pandora is a good way to start.
Go to KEXP.org and click on the the "Streaming Archive" link. They have weekly radio shows for most genres. For example, I regularly stream their reggae show "Positive Vibrations". You just click play and it will play the 3 hour show. Similarly, they have a blues show, a jazz show, etc. Just as KEXP streams there music, there are hundreds of others that do as well. In fact, the easiest way for you to start is by Googling whatever FM station you are hoping to listen to. You'll find that their internet stream is cleaner, might lack all commercials, and will give you info as far as song title and artist being played.
As far as exposure to new music, I subscribe to Rhapsody. For about $10 a month, I can stream anything I can dream up. I can also play Rhapsody through my iPhone and plug in into portable speakers or my car stereo. Essentially, Rhapsody gives me an unlimited, on-deman jukebox. The "downside" is the $10/month charge which I consider well worth the expense. For less than the cost of a CD, I can listen to dozens of new releases a month and find whatever I want. If a friend suggests a band, I can just search for it and check out all of there albums.
Getting your music from your computer to your speakers isn't particularly difficult once you know what to do. As Leland mentioned, the easiest way is to plug into the output on your computer's soundcard. There will be a "mini" plug port on the back of your computer(where you could plug in headphones or computer speakers). Go to Radio Shack and get a "(male) mini to (female) RCA connection". From there, run long RCA cables from that mini>RCA converter to an input on your home stereo. You now have your computer playing through your stereo. The downside of your particular case is that you'd have to go into your "computer room" to see what is playing and skip/select songs.
You could take it the next step fairly easily, which is integrating everything. For example, my computer is in my living room. The computer's audio output goes right to my stereo and the computer's monitor connects directly to my flat-screen TV. If you do this, you can do things like stream Netflix movies right to your good TV.
If I was in the area, I'd be happy to help. I'm guessing there are some other long-term PP members closer to you that wouldn't mind lending a hand if wanted some help?
-AP