Study what you already have before you start spending money to build from scratch. A shed in the backyard may be a good distance from the noisy street. Your garage can be a great choice if it has concrete walls.
Add layers of mass to the walls. This can be done by simply hanging up heavy carpet or blankets. Manufactured wall paneling or extra layers of sheet rock can do the trick. You can also hang foam acoustic tiles on the walls.
Eliminate vibrations from the outside by lining the stud faces with a pipe insulation made of a rubbery material.
Cover your floor with heavy carpet and a thick pad. Carpet also helps stop echoing. If you build a floating floor on top of the actual floor, you can add another layer of soundproofing.
Double glaze the windows. A lot of noise leaks into a room through the windows. If they are already double glazed, you can add another layer of double glazing. Make sure the panes are thermal insulated. Cover the windows with drapes made of heavy material. If you don't care about getting light through the windows, you should board them up and fill every space. If you are building a soundproof studio from the ground up, just leave out windows.
Add an interior door. No matter how thick your exterior door is, sound will leak around it. Putting in an interior door will help this, but you should also be careful to seal the spaces between the door and the walls, especially at the bottom of the door.