If you really want to do it right, it would be a big job. What you'd do is re-stud the walls so that the walls of your room don't share common studs with surrounding rooms. How? By staggering studs. Below is a very crude drawing of what I mean. the horizonatl lines are the walls to adjoining rooms,; the verticals are studs (the studs would actually be closer together inside the wall cavity, but I think you get the idea- keep vibrations from the room in question from travelling to other walls. Then fill between the studs with a thick insulation (ask your hardware supplier which is best rated for sound absorption.
Jon Bon Jovi built a music studio in his basement that was featured in Mix magazine a few years back. he wanted to soundproof it, and this is what he did. He even went so far as to detach the house from the foundation to totally isolate the basement.
If you're not into all of that, probably blow-in insulation will help. Again get the highest rated for sound absorption, and seal the doors and windows as much as possible, Drapes, carpets, etc. will also help.