Real Audio Player
If your looking to produce streaming audio, you should look into using Real Networks. While the technology is a little old, its certainly a good place to start learning the process of setting up streaming audio and video on any website.
The only downside of using Real Audio is that there isn't a Producer application for the Macintosh. They use to make a plug-in that worked with SoundEdit Pro. That's what I used when I first started putting audio on my website.
To get started you'll need the RealProducer, there's two version: RealProducer Plus and RealProducer Basic. Since your just starting out, I would recommend using the RealProducer Basic, its Free!
When linking to the RealAudio file, I would recommend using the rtsp:// format instead of http:// that you may be use to. Using rtsp:// forces the browser to open up RealAudio and play the audio as a stream media. Using the http:// will only download the music file and play it after the audio has been downloaded.
Example:
rtsp://www.cryan.com/realaudio/Concord.ra
Selecting the Right Audio
One of the things you should consider in picking your audio is what makes it unique from every other site that's out there. Why would people want to visit your webiste to hear your audio files? You need to find audio in your collection that is worth putting on your site.
Think PodCast!
The most popular audio file today isn't music, its Podcast. Podcasts are audio clips, usually about 5-10 minutes long talking about various topics. One of the advantages of Podcasting is sharing your expertise with other people. Think about a topic that you know very well that you could talk about
Copyright Laws
Just because you have a audio file doesn't mean you have the right to share it. You need to be aware of the copyright laws in your community to understand what you can put on your website and what's for your ears only. There's plenty of sites, in the United States the best site is Title 17 of the United States Copyright Laws.
Have fun
No matter what audio file you put on your website, have fun with the process and be creative. Think of some creative audio that isn't online yet and challenge yourself to do something different than what everyone else is doing.