Definition
To follow diligently.
To treat wrongly, especially in a romantic relationship; to insult or nag.
How People Actually Use It
To dog around is for a man who is in a relationship to out looking for ass, or other females willing to flirt, or flash a titty.
"The man leaves a voicemail*** "Hey, have fun doing what ever it is that you're doing. I'm going to do dog around after work. Peace out.""
Source: Urban Dictionary