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

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
                

b_limo

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

I’m starting to research ways to reduce sound transmission through a wall

If the common wall between you is "drywall’ on your side (we call it "Gyprock Sheet" in Australia).
You should remove it your side, put compressed "pink batts" (insulation) in the cavity all over, and then new "Dry Wall" nailed back over it to compress the pink batts in the cavity and paint the new wall. I held the batts in place with string thumb tacked into the wood battens while the new drywall sheets were nailed up.

Cost
"dry wall cheap as **** so are pink batts, and you don’t see anything, looks like it did.

Cheers George

I want to give you some suggestion which is very important for you. I think you can use some of noise barrier material which is most effective for you. I use insulated tarps for me and it is very nice working.

@b_limo   Appreciate your contribution. To have value the batts have to be cut just right to give a secure “press fit”. There will not be any need of support before adding drywall. Ensure the batts are slightly thicker than the studs. The you have the benefit of damping the drywall with the pressure of the batts. All this will pale in comparison to 5/8 Quietrock smply attached via green glue. No nails. If dabs of a different industrial glue are required  for strength, use one that remains resilient. This type of adhesive is readily available.