What is the most economical way to accomplish room soundproofing in your home?

The easiest and most cost effective way to soundproof an existing room is to add another layer of soundproofing drywall like Quietrock over the existing drywall - or replace. It's easy and now Lowe's is selling it for $34 per panel. It's very reliable even univeral studiOS uses it - you have to use materials designed specifically for high stc that won't get short-ciruited and fail when you hang picutres or anything like that on the walls

Edited to add:

Another great alternative is the use of Green Glue damping compound. It comes in a caulk-like container. All you have to do is squirt it over your existing walls, afix a new layer of drywall, seal the seams with acoustical caulk, and refinish the surface! It's easy and inexpensive!