One of my first memories of seeing standup comedy was on a family vacation. The hotel we were staying in had HBO, and Robin Williams was onstage. My parents overheard, and that was the end of that.

Nate Bargatze, Leanne Morgan and Dusty Slay are leading a family-friendly scene. Unlike earlier stand-ups, they don’t look down at their cursing peers. By Jason Zinoman Reporting from Nashville The ...