Place the two pieces of 30-by-30-by-3/4-inch plywood on a worktable. Apply wood glue to the top of one piece.
Lay the second piece on top of the first and align the sides. Secure it to the first with the 1 1/4-inch drywall screws. Clean up the excess glue with a damp cloth. This is the bottom of the utility cart.
Apply glue to the ends of the 2-by-6-by-30-inch boards. Secure them between the 2-by-6-by-33-inch boards with the 2 1/2-inch drywall screws. This is the frame of the utility cart. It fits around the bottom of the cart.
Put some glue around the edges of the bottom of the cart and slip the cart bottom inside the frame. Secure the frame to the bottom with the 2 1/2-inch drywall screws. Clean up excess glue as before.
Turn the cart over. Install the casters on the bottom of the cart, one in each corner. Turn the utility cart over.