You can do two things to eliminate noise between rooms.
One is to eliminate air borne sound leaks through outlets and light switches by caulking any holes. Tune an old radio between stations to generate white noise and place it in one room while you listen with a tube to your ear on the wall in the adjoining room. You will be able to locate sound leak sources. You can also add fiberglass or cellulose insulation between the walls. Make sure that heat registers don't provide a passageway for sound between the two rooms. This technique gets rid of high frequency noise, like water running.
The second technique is to add an additional layer of drywall to each side of the wall separating the two rooms. It is important to make sure that the layers of drywall are of different thickness on each side to prevent coupling. So, if you add 1/2" drywall on one side, add 5/8" on the other side, for instance. This gets rid of low frequency, structure borne noise like voices. There are also special springy wallboard hangers that may be used to isolate the wallboard.