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

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. 

Many years ago I purchased a pair of Dunlavy SC-IV demos from Soundex in PA. I was not impressed with the sound but I knew it was the room and not the speakers. I ended up buying acoustic ceiling tiles from a high school classmate who had a ceiling business. I used 48 2' x 2' tiles for a 16' x 20' room. The door to the room has a different 2' x 6' tile. The results were amazing and I still have them there today. I've seen them on Amazon. You can get them in either 2' x 2' or 2' x 8' sizes. They're about 1" thick and consist mostly of a yellow fiberglass material. The cover surface resembles black course sandpaper. They aren't pretty and I wouldn't recommend them for anything other than a dedicated listening room. Years after I installed them I was on a construction project called NFL films. One of the two buildings was all recording studios of different sizes. I was happy to see that they basically did the same thing I did with the damping on the walls. 
             Hope this helps,
                                    Frank
                

Have you also thought if just playing music QUIETER? I know it is not the answer, you seek... but if they are the owners...I live in an apartment. And I play music rather quietly all the time. I own Magnepan 20.7s. But I only play at about 50dB to 65dB most of the time. Hitting 73 or so now and then. Loud is a learned experience. You get very used to loud, and quiet sound terrible. (at first). But if all you do is play quiet. it begins to sound great. I used to many years ago play loud music. Now I do not disturb my neighbors.Another way is to go headphones. If I want louder, late at night. Headphones are the only way.