A term used in the 80's to define a male and a female who were an item, more than friends, were interested in each other exclusively, otherwise known as dating, or considered a couple. They were "going around" together.
When a boy was interested in a girl he would ask her to "go around" with him.
Sally and Joe are going around. Meaning they are an exclusive couple and have been seen around town together as a couple.
Pete asked Molly to go around with him. (Be seen in public as his girlfriend and date no other boy but him.)