There are a number of easy ways to get music. This is the official Ubuntu documentation, but it's a bit out of date. Everything there still works, but you have even more options now.
You can download music using your music player, your web browser, or file sharing software. Some of it is free, some costs money. Not all those options are legal everywhere. If you aren't sure you need to ask a lawyer, and not rely on opinions posted here.
The two most popular music players, Rhythmbox and Banshee, have several options built-in. Rhythmbox, the default audio player, has the Ubuntu One Music store, as well as Magnatune and Jamendo stores. Banshee also has the Magnatune and Jamendo stores, as well as plugins to download music from Amazon, eMusic, and Internet Archive. Ubuntu 12.10 is due to be released very soon and will probably have a new way to download music from Amazon. Not all these options will work in every country. You'll just have to try some and read the terms and conditions for each to learn more.
Using your web browser, you can download audio files just like any other file. Then add them to your music player if you want to keep them. Many bands release free sample tracks on their own sites, as well as music blogs. Try simple Google searches for types of music you like to see what's out there. This Ubuntu forums thread is a good starting point.
Depending on what country you are in, you can also just use a torrent tracker and/or Transmission (installed by default) to download music. Usenet is another popular option, which requires special software. These are the methods that are most likely to result in legal issues, so pay special attention to the laws of your nation if you try these. Another consideration is that these are unregulated and more likely to contain low-quality files and malware.
Whatever method you go with, just be sure to avoid files with Digital Restrictions Management. These are defective and frequently give users problems.