"If a pendulum goes around a circle in the horizontal plane, would the period be smaller or bigger than the period from when in simple harmonic motion?"

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So I'm a little confused on this, because the length of the pendulum isn't changing, and the value of "g" doesn't change, so I'm not sure how to determine how the period will change.

They are referring to what is called a conical pendulum in the first part. Such a pendulum does not swing back and forth. It moves in a circle at constant speed with the motion forming a cone in which the angle theta subtended by the string to the vertical is a constant.

It is difficult to explain without figures. Do look up the derivation of formula for the period and compare it with that of a simple pendulum.

For a conical pendulum, the period is given by

T = 2 pi SQRT (L cos theta / g).

For small theta cos theta ~ 1 so the period is the same.

As theta increases however, the preiod is smaller than that for a simple pendulum.