I started out with your budget. I ended up spending $5000 and a year of weekends and vacation days completing the project.
I first started by pulling off the existing walls and ceiling, which were plywood, and I put fiberglass insulation inside of them. I covered the walls with a layer of pressed sheath, which is available at home depot and is basically compressed cardboard. I then went over that with a layer of sheetrock. The only window/hole left was a door to the outside and a hole for a window unit. Next, on two sides of the shed I closed in areas for storage, a control room, and a foyer so that when you come in the front door, you are in a small area where you must open a second door to get to where the loud stuff actually happens.
After I did this i had a great big echo chamber. I put down some carpet, and made some bass traps and mid/high absorbers. This was the least expensive thing I did and the best money I spent because now the room sounds great.
Looking back, the most helpful thing I did was accidentally build it such that 50% of my walls in the loud room have a lot of dead space behind them. Those are the directions that kill the most sound. If I wanted to take another step and make it more soundproof, I'd go 6" away from the existing wall in the live room and build a room inside of that room, complete with ceiling, and follow the same process that I did with the existing garage. I am not this crazy however, and what i've done so far has worked great.
In summary, the project cost me 10 times what I expected at the onset and took 3 times longer than i expected to complete. However, the benefits of the room have far exceeded the toils needed to construct it. Every practice is at home, I can record whenever I want for free, and I can spend as much time working on music or sound or whatever and bother noone.
Good Luck in your efforts...
