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Step-by-step build — 5 steps. Also available as a print sheet.
Rip the plywood into three panels, each 15-3/4" wide x 48" long. Set a table-saw fence to 15-3/4" and run the sheet through, or measure, mark, and cut with a circular saw — or have the store make the cuts.
Assemble the two ladder-like leg frames from the 2x4 legs and shelf supports with 2-1/2" self-tapping screws (or pocket holes if you have a jig). Build them identical so the shelves land level.
Build the frame for each shelf first from the 1x3 ends and sides — nail and glue, use 2" screws, or pocket holes. Then apply glue to the bottom edges and nail the plywood deck onto each frame.
Set the shelves between the two leg frames at your chosen heights and screw them down, also screwing into the legs. Check the whole cart for square as you go.
Attach the four casters to the base, making sure all four sit flat so the cart doesn't rock. Finish with stain or paint if you like — bare is fine for a shop cart.