From the plug to the battery you probably want about 10 gauge. That's generally what is on the truck side anyway. The type of battery probably depends on how much light and how long you want them on. They make small gell cell batteries that are used for trailer breakaway switches and can be used for lighting. I'm not sure where you'd get one though. Generally you'd use a deep cycle RV battery.

You just need to find a place to put it, and put in a battery box. Run a 10 gauge wire from the plug to the battery. Run a 10 gauge ground wire from the plug to the frame, then from the frame to the battery. Then you can wire your lights to the battery however you see fit.