You don't need to change your Apple IDs region if you are just going on vacation. If you do want to change your Apple ID's region you would need to cancel your subscription first.
You can download Apple Music tracks for offline listening so you don't need to have access to Wi-Fi.
If you are going for an extended vacation i.e. more than a couple of weeks or so, you may need to let Apple Music connect to the servers for validation of your subscription every so often, so your downloads keep being valid. This can happen over cellular data and should use a small amount of data to validate.
See here: link-> Add music to iPhone and listen offline - Apple Support