Joint and plane all your lumber to make all faces flat and straight. This can be done with a hand plane or a jointer and planer.
Cut the skirt boards (the boards that hold the tops of the legs to the bottom of the top panel) with the table saw to the desired width and length. Remove the saw marks with the hand plane.
Cut the wood to create legs, lower skirt boards (the ones that will support the lower shelf) and the slats. Remove the saw marks with the hand plane.
Lay all the boards out for making one big panel that will make the top and the lower shelves. Be sure that their sides are straight or you will not get good joints. Apply glue to the sides of the boards and clamp them firmly, putting the clamps not more than 12 inches apart. Leave it to dry for at least one hour.
When the panel is dry, remove it from the clamps and, with a chisel or a handplane, remove any excess glue that has squeezed out from the joints.
Cut the top and the shelf to the desired dimensions with the table saw.
Cut tenons (projections) on the ends of the skirt boards and mortises (cavities) into the tops of the legs, and 5 or 6 inches above the bottoms of the legs where the lower skirt boards will go. You also need tenons on the ends of the slats, in the bottom of the top skirt board and the top of the bottom skirt board.
Glue together the slats, top and bottom skirt boards, and legs for the left side and the right side. After these assemblies are dry, they can be connected by gluing in the front and back skirt boards.
Finally, the lower shelf can be placed between the two lower skirt boards and screwed there by putting two screws through the skirt boards and into the ends of the shelf. The top is attached to the table by driving screws up through the bottom of the top skirt boards.