When I mute the iPad with the buttons on the side, the iPad is still playing the music quietly. This is really annoying and I would like to turn the sound completely off. Is there a solution to this?
iOS has alert volume, media volume, and a nasty bug.
By default, the physical volume buttons adjust the alert volume, and media volume can be adjusted in Control Centre opened by swiping down from the top-right of the display.
To use the volume buttons to adjust alert volume when no media is playing, and media volume when media is playing, go to Settings → Sounds → Change with Buttons.
If the volume is off but you can still hear media playing, this is a bug with iOS. If the media is playing full screen, you can mute the media with the volume button on the top-right of the media.