In general, what happens to the period of a spring and mass when you increase gravity?

The answer is nothing.

Explanation:

Unlike the pendulum made with a rod and a mass, the period of oscillation of this system is independent of gravity,

because the restorative force here is that produced by the shrinking and stretching of the spring, the weight of the mass is offset with an "initial tension" of the spring at the equilibrium position.

In a mass and rod system, the restorative force is a component of the weight of the mass itself, that's why the pendulum period for those cases is

where "l" is the length of the bar and "g" is the acceleration of gravity.