I'm having some trouble understanding circular motion. I know that centripetal force is equal to string tension, but should the two always equal the force of gravity?
not at all. In fact, gravity may not enter into it at all, if the circular motion is in a plane perpendicular to the gravitational force.
Now, if you are talking about two bodies, like the earth and moon, then of course, the only force acting on them is gravity. So, of course the centripetal force is exactly gravity. It is the "tension" keeping the moon in orbit.