Pour the concrete foundation. Use wood to mark off the area of the foundation and pour concrete in to create a solid floor for your recording shed. Allow to dry for a few days before continuing. Make the area small, at most, 10 feet by 10 feet. This will give you enough area to record a live drum set while also having the other musicians in the room.
Frame the walls all around the shed. You should use 2-inch by 4-inch wood for the framing, as you will need to insulate the shed well for soundproofing. Also place at least one door and possibly a cut-out for a wall-unit air conditioner, as it can get very hot in the summer.
Use the same framing wood to create a roof. Preferably, a pitched roof would be best, but you could do a flat roof, as it may be easier if you have limited carpentry skills. You want to place the framing about 18" apart from each other and possibly match the location of the room framing with the wall framing locations.
Use plywood to create the outside walls and roof. You can purchase plywood in large sheets that can be nailed directly to your framing. Since the framing studs are 18" apart, you can easily locate the area to be nailed to hold the plywood on securely.
Run electricity to the room by adding a small breaker box and run the wiring before attaching the drywall. You can cut out areas for the electrical sockets with a jigsaw. Measure the distance of the box from the floor and cut a hole in that piece of drywall so that it matches the location. You want to have at least three outlets in your shed.
Insulate inside the walls. Not only will this allow you to control the sound better, but it will help you regulate the temperature inside your shed. Use fiberglass insulation--as much as you can afford--in between each stud. You can also do this to the roof if you are ambitious and really want your shed to be insulated properly.
Create the walls inside your recording shed with drywall. Screw the drywall sheets to the studs in a similar fashion as the outside. For a professional look, putty the areas where the drywall pieces meet and paint the walls after you are done. You can also drywall the ceiling or add a drop ceiling for a completed look.