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I want to figure out why my rear light does not work. I understand electronic basics, here is my question: I have a power supply. Is there a place on the bike I can wire it in, so I can have the right voltage in the system, so I don't have to have the bike running to look for shorts and issues?

It's much easier to just use a multimeter and a couple questions...
1. Does your headlight work?
If yes, then unplug the tail light connectors (one is Black/Yellow and the other is Red). Connect your multimeter to the leads going TO the tail light and check continuity. If its open check your bulb to make sure it is good, if the bulb is good inspect the bulb socket for broken wires.
If that all i good then measure continuity on the Black Yellow wire going AWAY from the tail light to ground on the frame. IF open check your ground connection (it is up under the tank near the front of the bike.
If that is good, start the bike and check for voltage on the Red wire going AWAY from the tail light. It should be 12v or so. If not then your headlight also does not work so you need to check your lighting coil circuit. Disconnect the yellow wire from the light switch and check for voltage there. If good then check your light switch. If bad check the lighting coil (which is unfortunately under the flywheel and involves pulling the flywheel).

If you really want to use a power supply, then disconnect the yellow wire coming off the light switch and connect your 12v supply to the yellow wire going INTO the switch. Power it up and you should have lights. If not check all your connections, especially the grounds.

Yes, the front headlight does work. Thanks for the tips, I'll go down the multimeter path first. THANK YOU for the detailed info.

Since the headlight works it's likely 1 of 3 things....burned out bulb, broken socket, wires not plugged in fully on the bullet connectors. Sometimes they are a real pain to do. Since you're in there looking around just take a few minutes to clean up all your connections and add some dielectric grease to help with corrosion control.