It's actually not particularly complicated wiring - basic 1940s style car circuitry. But the headache is in trying to integrate it into a tractor that has a series of safety circuits. Some of them are really good ideas - like cutting out the PTO when you fall off the tractor and stuff like that. But there are a few that can cause trouble whenever you try to improvise and they may have to be bypassed. They can be complicated, as the whole idea is any one of them can trip and shut the tractor down.
Two automatic circuits are probably necessary for the motor to work predictably. The first is likely to be a hot wire to a fuel valve mounted on the carburetor - there's a little solenoid with a plunger, usually on the underside of the carburetor. When the power is off, the plunger (being spring loaded) extends like a finger - it's actually a needle valve - and interrupts the fuel flow. Its an automatic fuel shut off, initially designed as an anti backfire device so fuel wouldn't accumulate in the combustion chamber when the unit was off. It's not much of a problem with horizontal shaft engines but can be a real nuisance when the motor is on its side. This thing allows fuel to pass only when energized, so when current is cut off the engine will starve and stall.
The other automatic circuit, which may be your black wire, is called a "kill circuit." If you trace the black wire to the motor, it may end up attaching to the edge of the ignition coil or coils mounted adjacent to the flywheel. If it's a kill circuit, it will be mounted to the edge of the coil and not to the center. It will also be a small gauge wire. When the motor is shut off with the ignition switch, this wire is shorted to ground by the ignition switch. That in turn shorts out the coil(s) and prevents current from reaching the spark plug. Without this, the motor will continue to run. If you recall motors from many years ago, the kill switch was a metal strap attached to the head that you would press against the end of the spark plug to ground it and shut off the motor. If you have this wire, it has exactly the same function.
Briggs has a number of alternator designs, including unregulated designs. For instance, if you do not have a regulator, your red wire may be carrying DC 3 amps, unregulated, and your black wire 14 V AC, unregulated. There would be a diode in the connector to deliver DC current. So you should check out your motor's system to determine what you have. Member KennyV supplied this resource in another thread, which allows you to identify what system you have.
http://www.smallenginesuppliers.com/html/engine-specs/briggs/briggswiringexplanation.pdf
Your biggest amperage draw will be lights and your PTO clutch, which is likely to be a good sized electromagnet. The PTO clutch on my garden tractor is rated at 4 amps, meaning it has a 4 ampere current draw whenever it is engaged. That will be the case without regard to how hard the tractor is working. The amperage draw represents the energy needed for the PTO clutch to stay engaged, which is why you can engage and disengage the mower with a switch instead of using levers and belts and all that. Each of these can be wired to a switch or a relay - or you can use a Briggs ignition switch to do your switching for you.
Given that you have a new motor, I'd suggest not messing with its wiring. If it has a regulator, use it. That changes AC current from the stator to DC (rectifies it) and it absorbs excess voltage by shorting it to ground (regulator). With this thing operational, you will be producing an even current regardless of engine speed, so for instance your lights won't dim or burn too brightly - instead they stay steady.
One thing I had to take on faith at the beginning is that the motor will run all by itself - nothing on the tractor is needed for the thing to run. What the tractor brings to the motor's operation is the ability to control starting it and shutting it off. You need to track down a service manual for your unit. Try searching by serial number and check multiple resources - there's a lot of information here. You should be able to find pdf manuals to download both for you tractor and for the Briggs motor.
One other word - about the safety circuits: I have rolled my tractor twice. Once was when I was pulling a stump through a brook. I thought I had cut all the roots and as it turned out I had missed one the width of a pencil. It was enough and I found myself sitting in the brook with the tractor upside down on top of me in an instant. It immediately shut off because I was no longer sitting in the seat. Thank you, John Deere. The other time I was mowing a steep hill and did a slow wheelie backwards. This time I was again on my back and had the thing coming down with two very sharp blades looking for an ankle. I was able to kick with both legs and head the rig off to my right - it tumbled away from me. Again, the seat safety switch killed the motor so by the time the tractor actually hit the ground, the blades were probably no longer turning. Fortunately for me, it landed a few feet away from where I was, but still had the safety mechanisms not worked I might have been badly injured. Instead I felt appropriately humiliated, which I think is an excellent trade off.
So just because the safety systems can be bypassed is not a reason to bypass them. At the very least, make sure you keep the seat switch connected. If the tractor is old enough so you don't have one, consider installing one. People can get hurt making assumptions about this stuff.