There are plain circuit boards that can be customized. Its a plain board with copper on one side. With it, you can use a masking tape lining the circuit according to the schematic. Cut the masking tape to the desired width. Then use a ferric acid solution, soak the board for a few minutes. The uncovered part will be removed. And you will end up with a circuit. Drill holes for wathever components you will add. You can then solder the components on the circuit.
Note on ferric acid: use plastic container only to pour on!!! Be very careful, avoid contact with any part of the body.
Another option is a pre-drilled circuit board. The holes are lined up in a grid and will have copper sorrounding the holes. To complete the circuit, you can use copper wires (insulated to avoid short) to bridge other components.
I would go with the second option - inexpensive, the first one is a lot of hassle and dangerous and may not turn out as well as you wanted.
I don't know how much you know about the symbols, but at least learn the basics of the components. The schematic looks pretty simple and basic. Pick up a book. I can tell already that you need 24 resistors with different ohms, some switches a couple of batteries etc.
The white board is the circuit board. You don't need to have the exact same thing, as long as the flow is the same, the result is the same.