A plywood sheet, screwed and glued to the back side of the legs will make that as solid as a rock. The bottom shelf will also help with stability. Adding a 2X4" piece of wood along the back side will make a good back-splash. It will also give you an easy way to bolt the table to the wall. If you think the table will be disassembled in the future for transport, don't glue the top,back plywood, or shelf in place - just use screws.
Adding and edge of hardwood ( oak) to the edges will make the table look and last better, but isn't absolutely necessary. If your dad has a table saw, here is the best way to make the edges really robust. Take a 2X4 piece of wood ( oak would be perfect, but even pine will work). Set the blade at slightly more than the table top thickness, and set the fence 1" from the blade. Cut the length of the board. Now change the the blade height and fence so it will intersect the cut you just made. Cut the board, thus making an "L" shape that fits the top so it forms an edge. If it fits right, clamp and glue it into place. A #10X3" screw placed about every 12" down the edge will also be a good idea, and look good. Once all of these edges are installed, sand or plane the slightly raised lip down to flush with the table top. Run a finishing plane down the edges at a 45° to "break" the sharp edges. The simple addition of such and edge can make a hobby bench into a craftsman's bench. It also makes your bench top look 3.5" thick.
Note: If making a bench with raised lips to keep things from rolling off, mount the "L" so the short leg is on the top side, thus making a 3/4-1" raised lip. Do this along the ends and back side, with a flush edge on the front side. Most craftsman's benches are made this way.