I'll need the model, type and serial number off the engine to explain it all, but the starting circuit is pretty easy.
The solenoid will have one heavy cable from the positive on the battery and another to the post on the starter. The solenoid should have a spade connector on the side. You'll run a wire from the "start" position on the switch to this.
Will you be using the fuel shut-off solenoid on the carburetor? If so, that will need to be connected to the "run" position on the switch and maybe to the start position as well depending on the switch.
If you can give me the numbers off the engine I can explain the shut-off circuit. Lights generally require an AC source. Unless someone has removed it, there should be an alternator on this engine. It's located under the flywheel and there will be anywhere from one to four or five wires coming off it.
If this engine came off of a mower or other piece of equipment I can probably steer you to a diagram of that. Thanks, PK.
The following link is a very basic schematic that you can follow.
On your magneto there should just be one wire coming off it, off a spade connector. Is this not how it is? That would be your kill wire and would go to the off position where it would be grounded.
That diagram is for a Cub Cadet. It's the most basic diagram I have an online source for that I can give you the link to. The general schematic will work for any start/kill circuit and for lights and the charge system. Just ignore the safety switches unless you're using them.
On your engine specifically, we'll need to establish what the yellow, red and black wires are for. You'll need to pull the shroud and see which wire goes to the magneto kill and where the other two originate.
And yes, I got the numbers. I guess I was typing while you were.