The alternator produces alternating current. The battery provides direct current. Alternator current is rectified to direct current to charge the battery. To avoid over charging the battery the regulator shunts of excess voltage as heat. That is why regulators are finned or have a heat sink. Some times the regulator and rectifier are combined into one unit. There must be air flow over these components or heat will cause them to fail. You can't charge your battery with just the alternating current from the alternator. You must use a properly ventilated reg/rect assembly or it just won't work.
Since it doesn't matter to lights if they are run on ac or dc, some bikes will dedicate an alternator phase to lighting. Every other electrical component either requires DC, or will work better and last longer with DC. DC is used because the battery acts as a reservoir that provides stable output even under momentary high loads.
For proper, dependable wiring, run alternator output to a reg/rect properly mounted for cooling air flow. Run that to the battery as per the instruction that come with the reg/rect. I recommend a Rick's Electrics part # 10-515. On a large bike with large loads run a lead from the battery to a fuse box through the ignition switch, with a separate fuse to the ignition switch. Run everything from the fuse box, through the load (lights, ignition, etc.), then to ground. On a bike like the RE you probably don't need the second fuse box. One 20 amp should be enough for the loads you are likely be generating. You can run multiple loads off a circuit as long as you do not exceed the capacity of the wire to carry the load. You must calculate usage.
Run no more than a 20 amp fuse. Load should be at least 25% less than max. So a circuit with a 20A fuse should not be using more than 15 amps in normal use. A 10 amp circuit would have no more than a 7 1/2 amp load. The stock wiring is good for these ranges, but the wiring from the battery through the ignition switch to the fuse box must be able to take the load from ALL circuits. So it will usually be heavier duty than the rest of the wiring.
OOPS! If you have an early bike you will only use one phase, the two violet wires. The Rick's unit is wired for three phase. Just ignore one of the yellow wires on the Rick's unit.