How picky are you with carpeting? When we were planning to make a home theater room in the basement, we had already decided to lay that "boring" rough deck/patio carpet with "nubs" on the backside directly on the concrete floor. Breathes well, comfortable to walk on... (felt warm despite the uninsulated slab)... It was available in anthracite gray... (wanted a dark floor, but it comes in more colors)... we laid it in half of the basement and were super satisfied!
We had two ideas about the walls... One was to frame with metal studs... insulate and drywall... The other was to paint the wall and then hang fabric around the walls... a la molton fabric type... Behind the fabric, you can hang other sound-improving materials if you wish...
Unfortunately, we didn't finish before we moved :/
Good luck

Best regards, Sandra