Well, assuming you have a track loaded, place your mouse cursor at the beginning of where you want to start your cut. Make sure that you have the 'selection tool' selected. That will put a vertical line at where you want the audio to start. Then you can either click and drag your mouse cursor to the right to highlight the bits of audio you want to select. Alternatively, you can move your cursor to a location further to the right of the beginning and while pressing the shift key, click to about where you want the selection to end. You should see a grayed section of the audio 'selected'. This is your selection. You can now go into the File menu and click on 'Export selection as wav' and give it a name for your new sound clip. It will save it to an individual file with just that audio contained within.
There are all sorts of other tricks, but the simplest involve the selection tool and the shift+clicking. You can also shift+click-drag the grey part to as long of the audio that you want. Pay close attention to the timing that Audacity will tell you about with regards to where you are in the sound stream. To be precise you'll also want to zoom in/zoom out to really get where something starts or ends. This is especially important if you want to fix things, such as rebuffer errors and the like. After awhile you'll learn to recognize the wave-forms themselves and be able to 'see' individual audio before even playing it. For example I'm looking at a stream I recorded recently, and I can tell which parts of it are the songs, which are commercials, which are IDs, and where there was some silence; just by looking at the wave-form of the stream. I can see where I'm going to need to put a fade-in, or where a song ended abruptly and another was started. This comes from a couple of years of messing around with this. You'll eventually get it, it is not particularly difficult.
Other tools allow you to amplify the selection, fade-out, fade-in, and other things I've found that will help you get rid of DJ crossfades etc. It all just takes a little practice. Later you'll learn to copy selections and paste them into a new track so you can actually 'edit' tracks to get something completely different! Play around with the program, and I'm sure it will come to you. It took about six months for me to become really comfortable with editing audio, but once you know how to do it, you'll be able to do all that fancy stuff that the recording industry doesn't want you to know how to do.
