Why do I need internet to listen to my music library? I was taking a walk yesterday, and I was listening to music in my personal library. No Apple Music subscription. After awhile, my music stopped. I checked, and it said something about needing internet to access music and turning on cellular music in settings or something like that. It happened again today at the mall.  I'm confused... I thought that my personal library was on my device. Is it? By the way, I am using an iPod touch 6th generation and running (gulp) iOS 9 public beta. My best guess is it's the beta. Maybe they don't want all my music on my device in case I have to reset it. But wouldn't it be backed up to iTunes?
Yipee! I figured it out! My problem is, as I forgot to mention in the question, I just got a new iPod. Aaaaaaand, I never realized it, but apparently when you get a new device, you have to download all your music into your library. Otherwise, you can still listen to it, but only over internet. Probably to save space. Downloading your songs can be done by tapping on the song or album, and then tapping "Make available offline".

Thank you everyone for your help!