Which latitudes of earth do people experience the greatest tangential speed?

Tangential speed is dependent on an object's distance from the center of the circle. The farther away from the center an object is, the faster the object has to travel. Therefore, people living at a latitude of 0 degrees experience the greatest tangential speed.

The tangential speed is highest for people that live along the middle latitudes, a. k. a. the Equator.

This is because the Earth is rotating on it's axis (the poles) so the farthest latitudes move the farthest distance compared to higher latitudes.