Soundproof material for a dedicated room

If you are 'serious, serious' it is not time to fool around with any of the suggestions already made. Only lead wall paper will work. Yes, there is such a product, and it is expensive, with professional installation. Children? Forget it. I used this product over twenty years ago to 'silence' a furnace as part of a basement media room. Work well. Also had the duct work treated to cut down on air flow noise. As all have said, it will take trial and error to find the best solution. Your neighbors must be part of the solution.

If the wall cavities are empty, you could have an insulation company blow them full of cellulose. I did all the walls, and ceiling of my room done when I built. It actually helps alot.

Green glue between 2 levels of dry wall works well. A separate wall, decoupled from the first is an excellent idea as long as you do not triple leaf. Go to the website for The Sound Proofing Company and read their explanations in Sound Proofing 101. As Microstrip said, it is a complicated issue and many of the simple solutions (insulation between studs) really do not work very well, especially for bass.

I 2nd the green glue recommendation. I used to manage real estate and had a few tenants that used it with great success. It was a very cost effective solution. They suggest using 3 tubes per 4x8 sheet of 1/2 or preferably 5/8" thick drywall with lots of screws...basically sandwiching a new piece of sheetrock over the old wall with green glue between them to form a soundproof barrier. I think the website was greenglue.com....lots of testimonials on their website. I have no affiliation with the company whatsoever, it just worked out very well in the applications that I used it in...

Most of the above suggestions will help; but only help. In order to truly soundproof a room you have to build a "room inside a room"; IOW, isolate the source of the sound as much as possible. This is what recording studios do, and there is a lot of info available on line on the subject. If that is not possible, the most effective way to "help" is mass, mass, and more mass. The green glue/sheetrock suggestion is a good one. Then make sure all doors are of solid construction, and seal very well.