Measure and mark a 2-by-4 in equal, 2-foot increments. Place a carpenter's square on the edge of the board at each mark, and trace its edge with a pencil to make cutting lines. Cut the board at each line with a hand saw.
Arrange the boards on their sides, not flat, in the shape of a square frame. Butt the corners together. Insert two screws through both boards at each corner with a power drill, joining the corners. Make the frame sturdy; beauty is unimportant.
Stretch the hardware cloth across the top of the wood frame. Hardware cloth is strong, wire mesh. Cut it several inches wider than the frame, in all directions, using metal shears.
Staple one edge of the hardware cloth to one side of the frame. Stretch it tight across the top of the frame to the opposite side, and staple it to that board. Continue until all four sides of the hardware mesh are stapled to the frame. Add more staples around the frame to attach the mesh, leaving no loose gaps.
Set the frame across the top of the wheelbarrow.
Dig into the garden soil deep enough to remove pebbles that could interfere with plant growth. Typical vegetable root depth ranges from 1 1/2 feet to 6 feet deep, according to the University of Arizona. Turn the dirt over with the shovel, and loosen it as much as possible. Pick up clumps of grass with dirt attached, and tap them on the ground to remove the dirt.
Rake the soil with a heavy garden rake. If the soil is damp, use your hands to help break apart smaller pieces.
Move a shovelful of dirt from the ground to the top of the sifter. Shake and tap the sifter against the top edge of the wheelbarrow, letting the dirt fall through the mesh and into the wheelbarrow's reservoir. Drag a small, handheld shovel through the remaining material to help the rest of the dirt fall through the sifter.
Remove any earthworms from the screen and put them back into the garden. Earthworms help mix nutrients through soil. Dump the pebbles off the sifter into the empty bucket. Once the wheelbarrow is full of sifted soil, tip it forward and dump it out. Continue until you have sifted out all the pebbles you need to remove.