Said as self-congratulation, either to oneself or to draw others' attention to an accomplishment. Is usually harmless or joking, but can be obnoxious if done too much.
"I answered every question right. Go me!"
"I knew this would be the quickest way to get there -- go me!"
a phrase used to entice another person in order to fight them, usually is used as a joke and not an intensional fighting line.
Bob: hey how are you?
Jim: you dont even smoke!
Bob: GO ME!