What is the safest site I can go to to to download music to my computer to get onto a burnable disk?
If you want to pay for the music you download, iTunes store and beat port is probably the safest amongst all.
If you like a video on youtube, convert it and store it from http://www.video2mp3.net
After obtaining all the music you desire, store them in a folder and burn the folder to a disc through windows explorer
Bandcamp and Noisetrade sites lists some of the free music which you can download for free.
Makeuseof lists some of the best free music via Sound of Sunday form of article.
You can download from google play music
you can download from makeuseof
Copyright holders such as the MPAA, RIAA and BMI pretty much make downloading anything digital illegal, so I'm sorry but I can't make the recommendation that I'd like to make at this time.
Jeff, there was no mention of "free" or otherwise illegally downloading music. There are many sites where music can be downloaded legally, such as Amazon, SoundCloud, SoundOwl, mp3.com, makeloud, last.fm, and the Free Music Archive.
Jeff, your suggestion that downloading anything digital is illegal is false. If you buy digital music from a reputable source such as Amazon or iTunes then the artists and labels are getting paid their cut of revenues, hence not illegal. It's only illegal to download copyrighted music without paying for it through torrents and other outlets for piracy.
If you need royalty-free music you can download samples in : http://www.locutortv.es/musicalibredederechos.htm . It is no charge.