On web sites where you've 'clicked' and the computer has played the music file, try to use the right-hand mouse clicker, and depending on the browser you should see a menu.
It may be handy to know what browser you are using and whether you are using Mac or Windows (and which version of Windows, eg XP, Vista, Windows 7)... just to have a chance of giving the most likely options you will see.
Sometimes it might be 'save target as ...' or 'save link as ...'
That should allow you to save from the internet onto your computer hard drive. (afterwards you can copy to the MP3 player.)
When helping clients with a new Windows PC I set up a folder for internet items on the "C:" drive, and then another for items they download - it is possible to get more specific like MP3, software, etc, but at least the things they download can be saved away from the desktop, and not fill it up with random downloads... So they would have
c:\Internet and c:\Internet\Download