To go to various different parties in different parts of town, or basically party all night long.
The expression "paint the town red" has NOTHING to do with a girl on her period leaving spots of blood all over the town. (Where do people get this shit?)
Guy 1: "I'm bored as hell. I heard Jake invited a bunch of hot chicks to his party tonight?"
Guy 2: "Yeah. Let's go paint the town red, man."