How do I make messages on iPad stop interrupting my music
You do this by turning the iPad onto silent in a similar manner to iPhone. The difference is the side switch is configurable on iPad.

If you have the side switch set to "Lock Rotation" then:


Double tap the home button.
Swipe right.
Tap the sound icon to mute the device.


If you have the side switch set to "Mute" then:


Toggle the side switch like you do on iPhone.


You can change the side switch configuration in Settings.app -> General.