It definitely sounds like your connection and I can explain why:
The playing a couple of seconds before skipping is commonplace when:
- The music isn't set to 'Available Offline' and there's little to no internet connection
- Your Apple Music subscription is no longer valid
If when listening to the music you experience stoppage or stuttering, this would definitely suggest a bad connection.
Now although you believe your connection is good and I'm sure it is, the strength of the signal can vary. Try using Apple Music whilst connected to your Wi-Fi and in the same room as the router, I'm sure you will see fantastic results.
I, personally, have the same trouble as you and it's because there are many walls between my room and the router which reduce the signal strength.
The overall solutions would be: get a Wi-Fi repeater in your room to increase signal strength or set all your music to 'Available Offline' and sit near to your router while it downloads all the songs and then you should have no issues.
Hope this helps!