What are efficient ways to soundproof a room?

Double, Air-gapped Walls
This one was key, especially to keep outside sound out and instrument sounds in (like my drums)! Inside of the cinder-block garage, we built a room using 2×4’s (or maybe 2×6’s). We then built an ever-so-slightly smaller room inside of it, leaving about 4″ between the outer wall and the inner one. This air-gap helped block transmission of sound. Inside each wall we used fiberglass insulation.

Double Sheet Rock
On the walls mentioned above we used two sheets of 5/8″ sheet rock, not just one. This provided more mass, also good for blocking sound.

Foam Egg Crates, Foam Rubber, Carpet
We got lucky and found a dozen sheets of foam egg crate at a building auction and bought it for what one sheet of it would normally cost. We covered the walls in it. We used thick carpet on the floor, and some adhesive foam rubber sheets on the door and outside wall.

Adjacent Mixing tech room
We put the mixing consoles, PC, monitors, and other gear that generated noise in an adjacent room. If we were doing it “right” I probably would have wired up audio jacks to totally isolate the two rooms, but instead we just put a large conduit between the rooms to run the cabling. I stuffed the conduit full of rags to block the PC fan sounds.

Angled Window
We put a window between the studio and the mixing room so the tech could see the band and vice versa. To prevent the sound from going through, we used two panes of glass. The one on the mixing room side was mounted in a normal up-and-down position. The second pane, on the studio side was mounted at about a 45 degree angle, so sounds would be deflected down toward the floor. E.g. (Studio) \ | (Mixing Room).

AC outside of the studio
The studio got pretty hot with all that insulation, so we had to add a window AC unit.  Putting it in the studio was a no-go, so we mounted it in a window in the garage (outside the studio’s double walls) and then fashioned some ducting to run the cool air into the back corner of the studio. This kept the compressor noise out.

Treating the gear
Finally, we would pick up rattling from drum stands, clicks of guitar pedals, snare vibrations, etc. We fixed this as we went – things like foam, cardboard, electrical tape, and rags all helped.