There are a couple things to consider... You haven't mentioned how your house is configured. If you are hearing noise several rooms away from the source, you could have the noise travelling into a common attic space. I would check this out if it applies.
A lot of good possibilities have been provided thus far. But from my own experience, the best one to try would be blown in cellulose insulation. It has better acoustical properties than glass and is well within the skill set of the average home owner. I did my entire house with this in a day some years back.
As mentioned, it involved drilling holes in the drywall at the top and bottom of the stud cavity. The blower I used only required an 1-1/2 inch hole. You just blew it in the bottom till it came out the top. Since you are going for sound deadening as opposed to the insulating value, you can tape over the top opening and keep blowing just a bit to increase the density. This will improve the acoustical values and reduce settling. Then just patch and refinish the holes. You can do iit all from the garage side.
I would definitely try this before building new walls or modifying your existing ones.
P.S.If you elect to use this method, be sure all gaps are sealed before blowing the wall. It fills EVERY nook and cranny. I turned off the power to electrical boxes in the walls and then extended the switches and receptacles out so that I could stuff the boxes and prevent them from filling up.
Jim