What's a free app that lets you download music and listen to it without Wi-Fi?

I need an app that I can download music for free, then watch it later without Wi-Fi. Also, it would be great if it doesn't take up too much space on my phone. (I have an iPhone 5s)

I'm not aware of any app. Downloading copyrighted music for free is usually illegal in most countries, so we can't help you with that.

Your only options are:
1. Streaming services: Pandora, Spotify, Apple Music, etc.
2. ITunes on your computer or the app on your phone.

Why not just take your current mp3's, add to your itunes song list, then copy to your phone?

There's no app, you will need Wi-Fi to download it and if you need some good security app, you could get Leo Privacy Guard and gain utmost security.

NO such thing, sorry!

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's 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's 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.