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Step-by-step build — 5 steps. Also available as a print sheet.
Attach the two top rails and two bottom rails to the two 1x12 sides with 3/4" pocket holes and 1-1/4" pocket screws plus glue. If you'd rather not see pocket holes, drive 2" screws through the outside faces of the sides and countersink them.
Check the box for square across both diagonals before the glue grabs — everything downstream references off this box.
Set the three 1x12 shelves into the box — the top and bottom shelves land on top of and beneath the rails from the last step, the middle one wherever you want it. Pocket-screw them first, then reinforce by gluing and nailing into the 1x2 and 1x3 rails.
Fasten the two cleats to the inside face of the bookshelf with 1-1/4" screws and glue. Their top edge sets the table height, so measure their position up from the floor — the same number you'll use for the table legs. Get this wrong and the tabletop slopes.
Drill 3/4" pocket holes in both ends of the front apron and in one end of each side apron. Attach them to the two legs, flush to the inside faces, with 1-1/4" pocket screws and glue. Also drill a couple of pocket holes facing up in the aprons now — you'll use them to grab the tabletop.
Rest the free end of the table frame on the bookshelf cleats and screw the aprons to the cleats. Set the tabletop on the frame and drive screws up through the apron pocket holes into it.
Sand every edge and corner round — this is a toddler's table. Finish the top with a durable, wipeable topcoat; markers and glue live here now.