the cheapest way when I worked construction was doubling the drywall. Put up two sheets instead of one on all walls. Typical insulation in the walls was fine but double sheeting led to a significant reduction.

cool to hear - I have existing rooms that I replaced hollow core doors with solid core doors, but it didn't help much. I thought I would have to rip off gyprock to add better sound insulation, but maybe just adding another layer might help...

I assume it would only have to be on walls adjacent to other rooms, ie not exterior walls? And how did you really know this worked? Did you ever live in a room like that, or try this on someone who reported it helped back to you in some fashion? Thanks for the idea :D !

I worked on 2 houses and partially a 3rd where the buyer asked for a 'man cave' room. We'd put nicer insulation on all sides and slightly gap the board from the wall (hard to explain) then double layer the drywall. We'd use thicker board typically used on ceilings for all walls, and carpet the room (makes a difference), along with a solid door. A good window if it had a window, like double pane or a window marketed to be sound proof.

Then most importantly we'd go around the cracks of the room with a can of foam insulation and seal every hole we could find.

These rooms where pretty quiet. I can't say how much of a difference just the drywall would be, but if I wanted the cheapest way I'd do foam sealing and double drywall. Tearing everything out to replace behind wall insulation would only be something I'd do if there was zero insulation in the walls. I guess you could tell by knocking the walls with your hand. If they sound hollow you'll know. If you have a friend with a sound proofed room ask to walk around and get a feel for the space so you'll know what to aim for.

Drywall is easy to replace though and something more diy-ers should try more often imo!