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Step-by-step build — 9 steps. Also available as a print sheet.
Drill pocket holes in each end of the 15" 2x2 rails and screw and glue them to the 31.5" 2x3 uprights, forming two identical rectangular frames. The pocket holes end up on the inside of the changing table, so choose the uglier faces of your boards for this side.
Attach the long diagonal brace with 2" wood screws driven from the top and bottom, and glue. Screw and glue one short diagonal piece into the upper position with a screw placed slightly off-center where it will cross the long brace. Drill a pocket hole in the remaining short diagonal piece and attach it as the lower half of the second diagonal, using one wood screw and one pocket hole screw, so the two short pieces lap over and under the long brace where they cross.
Drill evenly spaced pocket holes in the three top-surface boards, choosing the uglier faces for the underside. Clamp the boards flat and screw and glue them together with the top-surface cleats to form one solid panel. A light sanding pass on top once assembled helps even out the joints.
Drill evenly spaced pocket holes in the three middle-shelf boards the same way. Clamp and screw and glue them together, sandwiched directly between the frame's 2x2 rail and 2x3 upright rather than resting on its own cleats.
Mark where the pocket holes should go so the backs of both shelves sit flush with the backs of the side frames. With the top rails flush with the outside top corners of the side frames, attach with 2" pocket hole screws, checking that both shelves sit square before driving the screws home.
Drill four pocket holes in the back rail (2x3) and attach the top panel with four 2" pocket hole screws through the back rail, plus 2" wood screws driven through the 2x2 cleats.
Cut the 1/4" plywood to 24" x 45-1/2" and nail or screw it to the back of the table. Drawing a line across the back first shows you exactly where to fasten.
Drill pocket holes at each end of the topper end pieces, adjusting your jig for 1" material, and placing the holes where they'll be best concealed. Assemble the topper with 1-1/4" pocket hole screws and glue. If staining, stain both the top panel and the topper before attaching the topper. Attach the finished topper to the top panel with 1-1/4" pocket hole screws.
Give every surface a final sanding and apply your stain or finish of choice.