Adding wall insulation can be expensive and above the skill of most do-it-yourselfers. In many older houses, particularly with masonry wall, there is little space inside the walls for additional insulation. It’s often worthwhile to spend more on insulation with the highest R-value per inch.
Any insulation you add to save energy, especially cellulose insulation that better fills the space between the walls, will help to soundproof somewhat. But you need additional improvements for significant noise dampening because sound travels mostly through the wall studs.
For your new bedroom, research the Sound Transmission Class rating. A typical uninsulated interior wall with drywall on each side of 2×4 framing has an STC of about 34. With an STC of 66, yelling is barely audible in adjacent rooms.
If there are common heating ducts and holes for electrical outlets and phone jacks, the STC may be only 25. These openings are also culprits for drafty rooms, so using an inexpensive outlet insulation kit also improves energy efficiency. Adding insulation inside the wall increases the STC by very little.
Your first step in soundproofing interior walls is to get out the caulk gun and seal gaps in walls and joints. It won’t help energy efficiency much, but it does block the vibrations that create sound.
Another consideration is the type of noise you want to block. For normal household sounds and voices, many standard soundproofing methods are effective. To block deep bass vibrations from music or a home theater, a thicker wall is best.
Decoupling the two surfaces of a wall is critical to block sound transmission. That means drywall on one side is not attached to the same wall studs as the drywall on the adjacent wall. One simple method is to install a second layer of drywall over the existing one. Make sure not to screw it into the wall studs or very tightly to the existing drywall so it stays decoupled. Another method is to stagger the studs on wider headers and footers in the wall cavity. The drywall on each wall is attached to every other stud so there is no direct path for the sound to travel.
Give windows the silent treatment
If your windows are old, installing new ones makes the greatest improvement in saving energy and blocking outdoor noise. Simply caulking and weatherstripping your old windows can have a dramatic effect on reducing noise, and it improves the energy efficiency of your house.