In my home, I had the guys who came to insulate put insulation between bedroom walls that shared the same wall and in the bathroom as well. They thought I was nuts as typically insulation only goes on the outside walls. But, I wanted to soundproof the rooms from one another. And, here we are, decades later and noises from one room to the other are vastly decreased by the insulation.
So, if you are redoing the walls, that is a great option that I KNOW works.
I would not put vinyl on a wall. That creates a place for moisture to collect, I presume.
You can get blown in insulation. They now make it out of recycled blue jeans, believe it or not. They place a half dollar size hole in the cavity of the wall between studs and blow in the insulation, then plug the holes for you to repaint.
Whatever you do probably depends on the money you have to put into the project, so do some price checking.