Best way(s) to soundproof interior wall between rooms?
The easy and fairly attractive solution is to use sound attenuation panels. You'll often see these in home theater settings. Hang them on the walls like you would a picture. You can DIY a similar solution by hanging a large thick tapestry on the wall, with an optional thin piece of foam behind it.

If you haven't already done it, carpeting in the room will make a huge difference.

To completely sound proof the room, you'd see the best results by removing the drywall from at least one side of each wall, and the ceiling if there's living space above, and installing specialized quiet drywall on top of sound attenuation channel that's attached to the studs like furring strips. This both separates the drywall from the studs to remove base sounds, and the quiet drywall absorbs the higher pitched sounds. However, even after all this work, there are still air ducts and doors that will often transmit sounds.