I can't seem to figure out these rigid body settings. A group of objects are parented to an empty, and I want the empty to bump into the ball and make it roll. I'd like that to be repeated a few times. I tried to put a rigid body constraint on the empty and select both the ball and the body of the figure (blue shape in red) but still nothing. When I do just put a rigid body on the blue shape and ball, they both just float away.
They are going to float away because they don't have a floor. Make a plane (size it to however big you want) and apply the rigid body to it, making it passive instead of active. Place the plane just below your objects.
After that, you should animate the blue object to hit the ball. Click "animated" in the rigid body settings for the blue object then animate it to hit the ball. You can animate the blue object to bump into the ball as many times as you want, but make sure that the ball stays on the plane. If you hit the ball too far, it will fall off the edge of the plane.
The ball and the "blue shape" should be set as rigid bodies, the box empty should not. They're currently being pulled downwards by the default gravity force; you can disable it in the "Scene Properties" tab.
With that out of the way, in the Physics tab set the "blue shape" as an animated rigid body by clicking the checkbox under the Settings submenu. Assuming the empty is already animated, it should give you the desired result.