Any apple music apps that let you downlaod free music and lets you listen to the music without wifi?

Unless you count any free songs through the iTunes Store, nope.

Yes. You already have one. The Music app that comes with your phone.

http://filebehind.com/APPLEAPPMUSIC

Please use LEGAL buy-a-download resources similar to iTunes or Amazon!

Please keep in mind that you can download/share only copyright-free material for free, unless the copyright owners allow free downloading/sharing of their material. ALL sites and resources that provide COPYRIGHTED material to be downloaded/shared for free, but WITHOUT permission from the copyright owners are equally ILLEGAL! So, unless you are about to download copyright-free material, please start using legal buy-a-download resources ONLY. Having no money/credit card is NEVER an excuse to STEAL anything anyway, so there is a very good reason why all the ILLEGAL sites/resources are being shut down and their users being sued for illegal downloading. It is also the reason why there is NO direct download button available on YouTube.

NOT all in music business are super rich, besides, all the studio engineers, backing vocalists, instrumentalists etc. DO deserve to get paid for the sheer hard work they do in order to make the music for us. I very much doubt YOU would work for 30+ years without getting paid either.