What's the best way to soundproof a room on budget

Will you own or rent your next dwelling? Vital info. Thx. Pete

@needfreestuff. Best suggestion. Cheers. Pete. Happy listening. 

Moving blankets (2 or 3 deep) on the walls are a good start

You can try  soundproof screen. Soundproof screen is suitable for new and existing structures, without the need for floor track. You can immediately customize each room to effectively use the space. With our soundproof dividers and screens, you can re-configure the rooms in the school, conference room, conference center and restaurant as needed.

I soundproofed a large walk in closet adjacent to my previous listening room (I've since moved), in order to reduce/eliminate the noise of the air compressor that drives my tonearm. Rather than ripping out dry wall and rebuilding the room, I used after market solutions- it wasn't easy, since any small opening will emit sound. I used wood furring (which functioned like bare studs), mass loaded vinyl (man, that stuff is HEAVY and it isn't cheap) and used sound blankets over the mass loaded vinyl (which has a hard surface and reflects sound- it is meant to be used as an underlayer). I also had to use weather stripping to seal the door. It worked - the biggest cost was the mass loaded vinyl material.
More recently, I had a silencer box built to contain the compressor- using marine grade plywood, two layers with green glue in between, and covered the interior with a specialty product - large sheets of melamine foam forming a sandwich around an inner layer of mass loaded vinyl. I had help from a competent carpenter to cut and assemble the thing-- it was based on an article from a sound proofing materials supplier about how to build a silencer box for an air compressor. Mine wound up being a little fancier than the simple box in the article-- a door to access the insides, various intake and exhaust fans, with a thermal controller and a few other refinements. It looks like Darth Vader's refrigerator. If you scroll down here, you can see the box in situ [url]https://thevinylpress.com/system-notes-austin-2017/[/url] Cost was high, labor varies depending on how much you can do yourself. It works. 
If a contractor can get a better price on the materials and knows what they are doing, it may offset the higher cost of materials and DIY labor. My experience in both instances was that there was more involved than I thought.